#6630692

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629009
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1886 conjugacy classes of subgroups; only the 570 normal subgroups are shown.

normal subgroups only

This group has 1886 conjugacy classes of subgroups — too many to draw. Only its 570 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 — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,d^b=di,e^c=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, 6537888);