#8115521

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	#34636	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4427 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,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, 8022717);