#10410011

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

4089 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,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, 10317207);