#8757321

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	#47479	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8664517);