#7638514

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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

5180 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)(11,13)(12,15)(14,16);(2,6)(7,8)(9,11,13,10)(12,15,16,14)(18,21)(22,24);(10,11)(14,15)(17,18)(19,22)(20,21)(23,24);(9,12,13,16)(10,14,11,15);(1,3,5,4)(2,7,6,8)(18,21)(22,24);(1,4,5,3)(2,8,6,7)(17,19,20,23)(18,22,21,24)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,e^c=ehi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7545710);