#10521306

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

4354 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=hi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10428502);