#10363722

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

4298 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,d^2=hi,e^2=g,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, 10270918);