#10464593

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

4091 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10371789);