#7665612

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#36576	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5296 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,15)(13,14)(18,21)(19,23);(9,11,14,12)(10,15,13,16)(18,21)(22,24);(2,6)(3,8)(9,10,14,13)(11,15,12,16)(17,18)(19,22)(20,21)(23,24);(17,19)(18,22)(20,23)(21,24);(1,3,5,8)(2,7,6,4)(9,12,14,11)(10,15,13,16);(1,4,5,7)(2,3,6,8)(9,13,14,10)(11,16,12,15)

Polycyclic presentation:

2^9|b^2=g,c^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7572808);