#8695880

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

4286 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603076);