#7671966

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

5948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 7579162);