#94812

order	512
nilpotency class	6
isoclinism	isoclinic class of #94810
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7504
commutator	C₄×C₈	C₄²
Frattini, cosocle	C₂×#492	C₂²
prime radical	C₂×#127	D₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#492 ◁ G
LECS	G ▷ C₂×#492 ▷ 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)(3,11,13,26)(4,15,17,25)(5,10)(6,9,19,23)(7,12,20,16)(8,18,24,32)(14,28,30,27)(21,31)(22,29);(1,3)(2,7,9,21,10,20,23,22)(4,16)(5,14)(6,13)(8,11,15,26,25,28,24,27)(12,18)(17,31)(19,30)(29,32)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=ghi,e^2=hi,f^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=egi,f^a=fg,g^a=gh,h^a=hi,c^b=ch,d^b=dfgi,f^b=fi,d^c=dgh,e^d=ehi,f^d=fhi,g^d=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, 2008);