#8115681

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
prime radical	#34643	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4337 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=gi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,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, 8022877);