#10247300

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

4006 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)(3,4)(5,13)(6,8)(7,15)(9,11)(10,14)(12,16)(17,18)(19,25)(20,29)(21,23)(22,26)(24,27)(28,30)(31,32)(33,34)(35,43)(36,41)(37,48)(38,44)(39,45)(40,47)(42,46);(1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,14)(15,16)(17,19,24,21)(18,25,27,23)(20,28,22,31)(26,32,29,30)(33,35)(34,40)(36,46)(37,44)(38,42)(39,47)(41,48)(43,45);(1,2,6,8)(3,9,11,4)(5,7,15,13)(10,14,16,12)(17,20,24,22)(18,26,27,29)(19,28,21,31)(23,30,25,32)(33,36,39,44)(34,41,45,38)(35,46,47,37)(40,48,43,42);(1,2)(3,9)(4,11)(5,7)(6,8)(10,14)(12,16)(13,15)(17,21,24,19)(18,23,27,25)(20,28,22,31)(26,32,29,30)(33,37)(34,42)(35,36)(38,43)(39,46)(40,41)(44,47)(45,48);(1,4,6,9)(2,3,8,11)(5,14,15,12)(7,16,13,10)(17,22,24,20)(18,26,27,29)(19,28,21,31)(23,32,25,30)(33,35)(34,43)(36,46)(37,44)(38,48)(39,47)(40,45)(41,42);(1,5)(2,7)(3,10)(4,12)(6,15)(8,13)(9,14)(11,16)(17,23)(18,19)(20,30)(21,27)(22,32)(24,25)(26,28)(29,31)(33,38,39,41)(34,44,45,36)(35,42,47,48)(37,43,46,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154496);