#8975202

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

4194 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)(3,10)(9,14)(11,16)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(33,34,38,36)(35,41,43,39)(37,44,46,40)(42,47,45,48);(1,2,6,7)(3,8,10,4)(5,9,13,14)(11,15,16,12)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(34,36)(39,41)(40,44)(47,48);(3,10)(4,8)(5,13)(9,14)(19,26)(20,24)(21,29)(23,30)(33,35,38,43)(34,39,36,41)(37,45,46,42)(40,48,44,47);(1,3,6,10)(2,8,7,4)(5,11,13,16)(9,15,14,12)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,4,6,8)(2,3,7,10)(5,12,13,15)(9,11,14,16)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,36,38,34)(35,39,43,41)(37,44,46,40)(42,47,45,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,37,38,46)(34,40,36,44)(35,42,43,45)(39,47,41,48)

Polycyclic presentation:

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

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, 8882398);