#7630506

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

11134 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)(6,7)(13,21)(17,22)(18,23)(19,24);(5,6,8,7)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24);(1,2)(3,4)(5,7,8,6)(10,15)(11,16)(17,22)(18,23);(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24);(9,13)(10,17)(11,18)(12,19)(14,21)(15,22)(16,23)(20,24)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537702);