#7633085

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

5720 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=h,e^2=g,f^2=i,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, 7540281);