#94796

order	512
nilpotency class	6
isoclinism	isoclinic class of #7529
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1408
commutator	C₄×C₈	C₄²
Frattini, cosocle	C₂×#494	C₂²
prime radical	C₂³	#353
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#494 ◁ G
LECS	G ▷ C₂×#494 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 206 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 36 points:

(5,6,8,14,11,16,24,31)(7,13,17,29,20,27,34,36)(9,15,23,32,10,25,22,30)(12,21,26,19,28,33,35,18);(1,2,3,4)(6,12,15,27,16,28,25,13)(14,29,30,35,31,36,32,26)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=gh,d^2=i,e^2=h,f^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=egh,f^a=fg,g^a=gh,h^a=hi,d^b=dfg,d^c=dgh,e^d=eh,f^d=fhi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1992);