#8621093

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=gh,e^2=i,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, 8528289);