#7629447

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5582 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536643);