#8695843

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

4171 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=h,f^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, 8603039);