#126954

order	512
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

720 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 202 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=hi,d^2=i,e^2=h,b^a=bd,d^a=dg,f^a=fi,g^a=gi,c^b=ci,e^b=egi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34150);