#8466983

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

6311 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8374179);