#94794

order	512
nilpotency class	6
isoclinism	isoclinic class of #94791
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7493
commutator	#127	C₄²
Frattini, cosocle	#3152	C₂²
prime radical	#532	C₂×C₄
LCS	G ▷ #127 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #3152 ◁ G
LECS	G ▷ #3152 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

179 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 179 conjugacy classes of subgroups — too many to draw. Only its 34 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 — 2 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=i,e^2=i,f^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=eh,f^a=fg,g^a=gh,h^a=hi,c^b=ch,d^b=dfhi,f^b=fh,g^b=ghi,d^c=dgh,f^c=fhi,g^c=gi,e^d=ei,f^d=fhi,g^d=gi

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1990);