#10521285

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

3980 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=hi,e^2=gi,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, 10428481);