#7631258

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

11448 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)(6,8)(9,10)(11,18)(12,16)(13,21)(14,15)(17,22)(19,20)(23,24);(5,6,7,8)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,24)(22,23);(1,2)(3,4)(10,15)(11,16)(17,21)(19,23);(9,10)(11,12)(13,17)(14,15)(16,18)(19,20)(21,22)(23,24);(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24);(9,13,14,22)(10,17,15,21)(11,19,16,23)(12,20,18,24)

Polycyclic presentation:

2^9|b^2=gi,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538454);