#7772016

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

6126 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=g,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7679212);