#8202265

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3846 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)(3,10)(4,7)(5,12)(6,8)(9,14)(11,15)(13,16)(17,18,23,27)(19,28,22,24)(20,26,21,25)(29,31,30,32)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(17,19,23,22)(18,24,27,28)(20,29,21,30)(25,31,26,32)(34,35)(39,41)(40,42)(46,47);(1,2,7,4)(3,10,6,8)(5,9,14,12)(11,16,13,15)(17,20,23,21)(18,25,27,26)(19,29,22,30)(24,31,28,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4,7,2)(3,8,6,10)(5,12,14,9)(11,15,13,16)(17,21,23,20)(18,26,27,25)(19,29,22,30)(24,31,28,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,32)(30,31);(1,6,7,3)(2,8,4,10)(5,13,14,11)(9,15,12,16)(17,22,23,19)(18,24,27,28)(20,29,21,30)(25,32,26,31)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8109461);