#8691035

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

4130 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8598231);