#7633133

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

6117 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^9|c^2=hi,d^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, 7540329);