#7631232

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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

6078 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=h,e^2=g,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, 7538428);