#93356

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#95
commutator	C₂×C₄	C₄×C₁₆
Frattini, cosocle	C₂²×C₄×C₈	C₂²
socle	C₂³	#336
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

344 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=fgh,h^2=i,b^a=bc,c^a=cf,e^a=efg,d^b=dg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 552);