#8845493

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

4041 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8752689);