#8196966

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

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

5779 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104162);