#7736956

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5263 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7644152);