#92816

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#99
commutator	C₂²×C₄	C₂×C₁₆
Frattini, cosocle	C₂×#336	C₂²
socle	C₂³	#323
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂×#336 ▷ C₂³×C₄ ▷ C₂² ▷ C₁

286 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2)(3,5)(4,7)(6,8)(9,10)(11,17)(12,20,21,15)(13,16)(14,18,22,24)(19,23)(25,26,27,30,28,31,33,36,29,32,34,37,35,38,39,40);(2,5)(3,8)(6,7)(9,11,13,19)(10,14,16,22)(12,18,21,24)(15,17,20,23)

Polycyclic presentation:

2^9|a^2=d,b^2=h,c^2=f,d^2=g,g^2=i,b^a=bc,c^a=ce,e^a=eh,c^b=cf,d^b=def,d^c=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12);