#8757592

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

4478 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664788);