#7670722

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

5130 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=h,e^2=g,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, 7577918);