#10407781

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

3949 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=ghi,e^2=hi,f^2=hi,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, 10314977);