#10340251

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

4159 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:

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

Polycyclic presentation:

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

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, 10247447);