#7631007

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

5814 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)(10,14)(15,16)(18,22)(23,24);(2,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18,21,22)(19,23,20,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19,21,20)(18,23,22,24);(2,5)(6,8)(9,11,13,12)(10,15,14,16);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,16,14,15)(17,20,21,19)(18,24,22,23);(1,3)(2,6)(4,7)(5,8)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=h,e^2=g,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, 7538203);