#7934522

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

5813 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=gi,c^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7841718);