#93578

order	512
nilpotency class	4
isoclinism	isoclinic class of #93517
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#340
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂²×C₄×C₈	C₂²
prime radical	C₂⁴	#103
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

310 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efghi,c^b=cg,d^b=dh,g^b=gi,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 774);