#8260665

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

4134 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)(3,8,14,9)(4,15,5,6)(11,13,12,16)(17,18,24,28)(19,29,31,25)(20,22,21,26)(23,27,30,32)(33,34)(35,42)(36,41)(37,38)(39,43)(40,46)(44,45)(47,48);(1,3,7,14)(2,8,10,9)(4,12,5,11)(6,13,15,16)(17,19)(18,25)(20,27)(21,32)(22,23)(24,31)(26,30)(28,29)(33,35)(34,40)(36,44)(37,45)(38,47)(39,46)(41,48)(42,43);(1,4,7,5)(2,6,10,15)(3,11,14,12)(8,16,9,13)(17,20,24,21)(18,26,28,22)(19,27,31,32)(23,25,30,29)(33,36,39,38)(34,41,43,37)(35,44,46,47)(40,48,42,45);(1,5,7,4)(2,6,10,15)(3,12,14,11)(8,16,9,13)(17,21,24,20)(18,22,28,26)(19,27,31,32)(23,25,30,29)(33,37)(34,36)(35,45)(38,43)(39,41)(40,44)(42,47)(46,48);(1,3)(2,9)(4,11)(5,12)(6,13)(7,14)(8,10)(15,16)(17,22)(18,20)(19,30)(21,28)(23,31)(24,26)(25,32)(27,29)(33,38,39,36)(34,37,43,41)(35,44,46,47)(40,48,42,45);(1,6)(2,4)(3,13)(5,10)(7,15)(8,12)(9,11)(14,16)(17,23,24,30)(18,27,28,32)(19,22,31,26)(20,29,21,25)(33,35)(34,42)(36,47)(37,45)(38,44)(39,46)(40,43)(41,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=h,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8167861);