#7632460

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

10980 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)(11,20)(12,17)(13,18)(14,16)(15,19)(21,24)(22,23);(5,6,7,8)(9,11,15,14)(10,16,19,20)(12,22,13,21)(17,24,18,23);(1,2)(3,4)(9,12,15,13)(10,17,19,18)(11,21,14,22)(16,23,20,24);(9,13,15,12)(10,18,19,17)(11,21,14,22)(16,23,20,24);(9,14,15,11)(10,16,19,20)(12,22,13,21)(17,23,18,24);(9,10)(11,20)(12,17)(13,18)(14,16)(15,19)(21,24)(22,23)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=i,e^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, 7539656);