#8095820

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

5335 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:

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

Polycyclic presentation:

2^9|a^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003016);