#8417237

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

3977 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)(3,8,13,15)(4,5,10,9)(11,16,14,12)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34)(35,39)(36,42)(37,43)(38,41)(40,44)(45,48)(46,47);(2,7)(4,10)(8,15)(11,14)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,35)(34,41)(36,45)(37,46)(38,40)(39,44)(42,47)(43,48);(2,7)(5,9)(8,15)(12,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,36,40,37)(34,42,44,43)(35,45,38,46)(39,48,41,47);(1,3,6,13)(2,8,7,15)(4,14,10,11)(5,12,9,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,37,40,36)(34,43,44,42)(35,45,38,46)(39,48,41,47);(1,4,6,10)(2,9,7,5)(3,11,13,14)(8,12,15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,38,40,35)(34,41,44,39)(36,46,37,45)(42,47,43,48);(1,5,6,9)(2,10,7,4)(3,12,13,16)(8,11,15,14)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,39,40,41)(34,35,44,38)(36,47,37,48)(42,46,43,45)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8324433);