#7632964

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

5519 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540160);