#8093232

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

4130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(2,6)(3,4)(10,13)(11,12)(18,23)(21,31)(24,30)(25,32)(36,43)(40,45)(42,47)(46,48);(1,2,7,6)(3,9,4,8)(5,13,14,10)(11,15,12,16)(19,27)(20,29)(24,30)(25,32)(34,38)(35,41)(40,45)(42,47);(1,3,7,4)(2,8,6,9)(5,11,14,12)(10,15,13,16)(17,18,22,23)(19,24,27,30)(20,25,29,32)(21,26,31,28)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,4,7,3)(2,9,6,8)(5,12,14,11)(10,16,13,15)(17,19)(18,24)(20,28)(21,25)(22,27)(23,30)(26,29)(31,32);(1,5,7,14)(2,10,6,13)(3,11,4,12)(8,15,9,16)(17,20,22,29)(18,25,23,32)(19,28,27,26)(21,30,31,24)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,6,7,2)(3,9,4,8)(5,13,14,10)(11,16,12,15)(17,21)(18,26)(19,25)(20,30)(22,31)(23,28)(24,29)(27,32)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

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

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, 8000428);