#99176

order	512
nilpotency class	3
isoclinism	isoclinic class of #8760
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

592 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=d,e^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fg,e^c=eh,f^c=fg

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372);