#8417017

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
prime radical	#53902	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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)(3,13)(4,6)(5,10)(7,11)(8,12)(9,16)(14,15)(17,18,22,26)(19,27,30,23)(20,31,29,24)(21,25,28,32)(33,34,38,39)(35,36,41,44)(37,40,45,46)(42,43,47,48);(1,3,8,16)(2,9,12,13)(4,14,5,11)(6,7,10,15)(17,19,22,30)(18,23,26,27)(20,21,29,28)(24,25,31,32)(34,39)(35,41)(40,46)(42,47);(1,4,8,5)(2,10,12,6)(3,14,16,11)(7,9,15,13)(17,20,22,29)(18,24,26,31)(19,28,30,21)(23,32,27,25)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,47)(46,48);(1,5,8,4)(2,10,12,6)(3,11,16,14)(7,9,15,13)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,27)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,6,8,10)(2,5,12,4)(3,15,16,7)(9,14,13,11)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,32)(29,31);(1,7,8,15)(2,11,12,14)(3,10,16,6)(4,13,5,9)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,31,27,24)(33,37,38,45)(34,40,39,46)(35,42,41,47)(36,43,44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,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, 8324213);