#7632229

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

10870 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,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, 7539425);