#7768136

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

5415 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=gi,e^2=g,f^2=g,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, 7675332);