#129342

order	512
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1064 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1064 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2)(3,4)(5,6,8,11,9,12,7,10)(13,14)(15,21,22,28)(16,24,26,19)(17,20)(18,23)(25,27);(13,15,17,22)(14,18,20,27)(16,25,26,23)(19,28,24,21);(2,4)(5,7,9,8)(6,10,12,11)(15,22)(18,27)(21,28)(23,25)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=di,f^b=fgi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36538);