#152736

order	512
nilpotency class	5
isoclinism	isoclinic class of #152734
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13088
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
prime radical	#3500	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3500 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

566 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,8,15,30)(4,19,20,38)(5,11)(6,12,27,10)(7,13)(9,21,33,24)(14,41,42,62)(16,29,46,59)(17,34,18,31)(22,50)(23,37,48,60)(25,39,54,36)(26,40)(28,55,57,47)(32,53)(35,52)(43,56,64,58)(44,49,45,63)(51,61);(1,3,7,18)(2,8,10,34)(4,14,20,44)(5,23,26,16)(6,15)(9,28,35,57)(11,37,36,59)(12,30,13,31)(17,27)(19,49,61,63)(21,47,24,43)(22,45)(25,48)(29,39,60,40)(32,58,33,56)(38,41,50,62)(42,51)(46,54)(52,64)(53,55);(1,4)(2,9)(3,14)(5,24)(6,22)(7,20)(8,28)(10,35)(11,38)(12,32)(13,33)(15,45)(16,47)(17,42)(18,44)(19,40)(21,26)(23,43)(25,53)(27,51)(29,49)(30,58)(31,56)(34,57)(36,50)(37,41)(39,61)(46,64)(48,55)(52,54)(59,62)(60,63)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,d^2=hi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dgh,e^b=ei,f^b=fgh,g^b=ghi,h^b=hi,d^c=di,e^c=ei,f^c=fi,f^d=fhi,g^f=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286987, #72287238
maximal normal subgroup of	#72286987, #72287238

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59932);