#10424607

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

4264 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:

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

Polycyclic presentation:

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

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, 10331803);