#8975375

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8882571);