#8630919

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	#56606	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4197 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

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, 8538115);