#10328335

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
prime radical	#53163	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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)(3,9)(4,8)(5,6)(7,10)(11,16)(12,14)(13,15)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(34,40)(35,36)(38,42)(44,46);(2,6)(3,10)(8,14)(11,13)(19,27)(21,31)(23,32)(25,28)(33,34,39,40)(35,43,36,41)(37,42,48,38)(44,45,46,47);(4,12)(8,14)(11,13)(15,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,35,39,36)(34,41,40,43)(37,44,48,46)(38,45,42,47);(1,3)(2,7)(4,13)(5,10)(6,9)(8,16)(11,12)(14,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,36,39,35)(34,41,40,43)(37,44,48,46)(38,47,42,45);(1,4,5,12)(2,8,6,14)(3,11,10,13)(7,15,9,16)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,37)(34,42)(35,44)(36,46)(38,40)(39,48)(41,47)(43,45);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)(33,38,39,42)(34,37,40,48)(35,45,36,47)(41,44,43,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=h,e^2=gh,f^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, 10235531);