#7684745

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

5100 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=g,e^2=hi,f^2=gi,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7591941);