#10519362

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

4051 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,e^2=ghi,f^2=gh,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, 10426558);