#8407079

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	#55700	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3852 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18,24,28)(19,29,23,25)(20,31,30,26)(21,32,27,22)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(1,3,7,12)(2,8,11,16)(4,15,9,6)(5,10,14,13)(17,19,24,23)(18,25,28,29)(20,27,30,21)(22,31,32,26)(34,39)(35,40)(41,46)(42,47);(3,12)(6,15)(8,16)(10,13)(17,20)(18,26)(19,21)(22,29)(23,27)(24,30)(25,32)(28,31)(33,35,38,40)(34,36,39,44)(37,42,45,47)(41,43,46,48);(1,4)(2,5)(3,6)(7,9)(8,13)(10,16)(11,14)(12,15)(17,21,24,27)(18,22,28,32)(19,30,23,20)(25,26,29,31)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,46)(45,48);(1,5)(2,9)(3,13)(4,11)(6,16)(7,14)(8,15)(10,12)(17,22)(18,27)(19,31)(20,29)(21,28)(23,26)(24,32)(25,30);(1,6)(2,10)(3,4)(5,16)(7,15)(8,14)(9,12)(11,13)(17,23,24,19)(18,25,28,29)(20,27,30,21)(22,26,32,31)(33,37,38,45)(34,41,39,46)(35,42,40,47)(36,43,44,48)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=h,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, 8314275);