#150089

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂×#41	C₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

219 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 219 conjugacy classes of subgroups — too many to draw. Only its 92 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=fi,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,d^a=df,d^b=dfi,e^b=ei

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, 57285);