#94452

order	512
nilpotency class	5
isoclinism	isoclinic class of #94449
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7313
commutator	#117	C₄²
Frattini, cosocle	#3509	C₂²
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ #117 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ #3509 ◁ G
LECS	G ▷ #3509 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

262 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 38 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,d^2=i,e^2=i,f^2=i,b^a=bc,c^a=cf,e^a=egh,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,d^b=dfh,f^b=fh,h^b=hi,d^c=dhi,g^c=gi,e^d=eh,f^d=fi,g^d=gi,f^e=fi

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, 1648);