#7632364

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

11328 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)(9,10,16,14)(11,17,20,15)(12,21,13,18)(19,23,22,24);(5,6)(7,8)(9,11)(10,17)(12,22)(13,19)(14,15)(16,20)(18,23)(21,24);(1,2,3,4)(9,12,16,13)(10,18,14,21)(11,19,20,22)(15,23,17,24);(9,13,16,12)(10,18,14,21)(11,19,20,22)(15,24,17,23);(9,14,16,10)(11,15,20,17)(12,21,13,18)(19,23,22,24);(9,15)(10,11)(12,23)(13,24)(14,20)(16,17)(18,19)(21,22)

Polycyclic presentation:

2^9|a^2=i,c^2=h,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, 7539560);