#7633293

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

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

11372 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,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, 7540489);