#150048

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

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	C₂²×D₄	C₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

270 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,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, 57244);