#153198

order	512
nilpotency class	5
isoclinism	isoclinic class of #153197
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13348
commutator, Frattini, cosocle	#532	C₂³
prime radical	#3018	C₂²
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

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

normal subgroups only

This group has 480 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)(3,14)(4,20,24,41)(5,25)(6,11)(7,29)(8,13)(9,16)(10,22,39,27)(12,31)(15,32,48,62)(17,34)(18,52)(19,40)(21,37,57,38)(23,56,30,36)(26,42)(28,58)(33,51)(35,53,61,47)(43,46,60,50)(44,54)(45,55,64,63)(49,59);(1,3,8,19)(2,9,13,34)(4,15,24,48)(5,17,6,16)(7,18,31,54)(10,35,39,61)(11,40,25,14)(12,33,29,59)(20,55,41,63)(21,45,57,64)(22,50,27,46)(23,53,30,47)(26,51,28,49)(32,38,62,37)(36,60,56,43)(42,52,58,44);(1,4)(2,10)(3,15)(5,22)(6,27)(7,30)(8,24)(9,32)(11,41)(12,37)(13,39)(14,43)(16,46)(17,50)(18,53)(19,48)(20,25)(21,26)(23,31)(28,57)(29,38)(33,35)(34,62)(36,42)(40,60)(44,55)(45,49)(47,54)(51,64)(52,63)(56,58)(59,61)

Polycyclic presentation:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60394);