#153174

order	512
nilpotency class	5
isoclinism	isoclinic class of #153173
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13346
commutator, Frattini, cosocle	#532	C₂³
prime radical	#33564	C₂
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

544 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,7,12)(3,8,17,36)(4,18,23,38)(5,24,26,10)(6,27,30,44)(9,20,40,21)(11,31,43,25)(13,45,48,61)(14,35,16,33)(15,50,28,60)(19,55,51,37)(22,39,29,42)(32,47,62,46)(34,49,41,52)(53,57,63,56)(54,59,64,58);(1,3)(2,8)(4,19)(5,14)(6,28)(7,17)(9,37)(10,33)(11,41)(12,36)(13,22)(15,30)(16,26)(18,53)(20,57)(21,56)(23,51)(24,35)(25,52)(27,50)(29,48)(31,49)(32,39)(34,43)(38,63)(40,55)(42,62)(44,60)(45,54)(46,59)(47,58)(61,64);(1,4)(2,9)(3,13)(5,20)(6,29)(7,23)(8,32)(10,38)(11,42)(12,40)(14,47)(15,51)(16,46)(17,48)(18,24)(19,28)(21,26)(22,30)(25,59)(27,54)(31,58)(33,61)(34,55)(35,45)(36,62)(37,41)(39,43)(44,64)(49,56)(50,63)(52,57)(53,60)

Polycyclic presentation:

2^9|a^2=i,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dhi,e^a=ei,f^a=fgh,g^a=gi,c^b=cf,d^b=dhi,f^b=fi,g^b=gh,d^c=dg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fh,g^f=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209895
maximal normal subgroup of	#72209895
maximal quotient of	#72227849, #72228014, #72228037

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60370);