#7627707

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄³	C₂³
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₄³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6788 conjugacy classes of subgroups; only the 3218 normal subgroups are shown.

normal subgroups only

This group has 6788 conjugacy classes of subgroups — too many to draw. Only its 3218 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) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^9|d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,c^b=ci

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534903);