#94381

order	512
nilpotency class	5
isoclinism	isoclinic class of #7396
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#353
commutator	C₂×M₁₆	C₄²
Frattini, cosocle	#2929	C₂²
socle	C₂	#7283
prime radical	#525	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2929 ◁ G
LECS	G ▷ #2929 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

236 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,4,10,5,12,17,27,7,13,19,28,6,11,18,26)(3,8,15,21,14,9,24,25,16,23,29,32,20,22,30,31);(2,8)(3,14,16,20)(9,12)(10,21)(11,22)(13,23)(15,24,29,30)(25,27)(26,31)(28,32)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=fh,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gh,e^c=ehi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1577);