#7633127

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

5516 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,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, 7540323);