#7652836

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

5343 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)(3,6)(4,7)(5,8)(10,14)(15,16)(17,18)(19,21)(20,23)(22,24);(9,10)(11,15)(12,16)(13,14)(18,19)(22,23);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,15,14,16)(17,19,21,18)(20,23,24,22);(1,3)(2,5)(4,8)(6,7)(17,20)(18,22)(19,23)(21,24);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,15,14,16);(1,2)(3,5)(4,7)(6,8)(9,12,13,11)(10,16,14,15)

Polycyclic presentation:

2^9|c^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

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, 7560032);