#94060

order	512
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₄×C₃₂ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂²⋊C₄
commutator	C₂²	C₄×C₃₂
Frattini, cosocle	C₂³×C₁₆	C₂²
socle	C₂³	C₂²⋊C₁₆
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

254 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 44 points:

(5,6)(7,11,12,9)(8,10)(13,14,15,19,16,20,23,29,17,21,24,30,26,32,35,39,18,22,25,31,27,33,36,40,28,34,37,41,38,42,43,44);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=g,g^2=h,h^2=i,b^a=bc,c^a=cf,d^b=df

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1256);