#10363737

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

4166 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)(18,23)(19,25)(21,29)(28,31)(33,34)(35,40)(36,43)(37,41)(38,39)(42,47)(44,48)(45,46);(3,4)(6,8)(11,12)(13,15)(17,18,22,23)(19,20,25,24)(21,26,29,30)(27,28,32,31)(34,39)(36,40)(37,45)(42,48);(1,3,7,4)(2,8,10,6)(5,12,14,11)(9,13,16,15)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,35,38,43)(34,40,39,36)(37,42,45,48)(41,47,46,44);(1,4,7,3)(2,6,10,8)(5,11,14,12)(9,15,16,13)(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,32)(30,31)(33,36)(34,35)(37,44)(38,40)(39,43)(41,42)(45,47)(46,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,20)(18,25)(19,23)(21,28)(22,24)(26,32)(27,30)(29,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,6)(2,3)(4,10)(5,13)(7,8)(9,11)(12,16)(14,15)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)(33,37,38,45)(34,41,39,46)(35,42,43,48)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=hi,f^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, 10270933);