#10360935

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

4270 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:

(1,2,7,4)(3,5,8,11)(6,9,15,12)(10,13,14,16)(17,18)(19,25)(20,22)(21,24)(23,27)(26,31)(28,29)(30,32)(34,40)(35,36)(38,42)(44,46);(1,3,7,8)(2,5,4,11)(6,14,15,10)(9,16,12,13)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,34,39,40)(35,43,36,41)(37,42,48,38)(44,45,46,47);(3,8)(5,11)(10,14)(13,16)(21,29)(24,28)(26,32)(30,31)(33,35,39,36)(34,41,40,43)(37,44,48,46)(38,45,42,47);(1,4,7,2)(3,5,8,11)(6,12,15,9)(10,13,14,16)(17,20,22,18)(19,23,27,25)(21,24,29,28)(26,31,32,30)(33,36,39,35)(34,41,40,43)(37,44,48,46)(38,47,42,45);(1,5)(2,8)(3,4)(6,13)(7,11)(9,14)(10,12)(15,16)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,37)(34,42)(35,44)(36,46)(38,40)(39,48)(41,47)(43,45);(1,6)(2,9)(3,10)(4,12)(5,13)(7,15)(8,14)(11,16)(17,19)(18,25)(20,23)(21,26)(22,27)(24,31)(28,30)(29,32)(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=ghi,c^2=h,d^2=ghi,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, 10268131);