#8109989

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

5350 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

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, 8017185);