#94039

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 170 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) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=fg,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=ef,c^b=ci,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1235);