#8458604

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

4109 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8365800);