#94776

order	512
nilpotency class	6
isoclinism	isoclinic class of #7519
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1402
commutator	C₄⋊Q₈	C₄²
Frattini, prime radical, cosocle	#3147	C₂²
socle	C₂	#7485
LCS	G ▷ C₄⋊Q₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #3147 ◁ G
LECS	G ▷ #3147 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

199 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 36 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1972);