#8108329

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

6008 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=gh,e^2=h,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, 8015525);