#8588395

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	#55491	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4043 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)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,18,23,26)(19,24,27,21)(20,22,25,31)(28,29,32,30)(33,34,39,44)(35,45,46,40)(36,42,37,41)(38,43,48,47);(1,3,7,9)(2,4,5,12)(6,10,15,16)(8,11,13,14)(17,19)(18,24)(20,28)(21,26)(22,29)(23,27)(25,32)(30,31)(33,35,39,46)(34,40,44,45)(36,43,37,47)(38,41,48,42);(2,5)(3,9)(8,13)(10,16)(19,27)(21,24)(28,32)(29,30)(33,36,39,37)(34,41,44,42)(35,43,46,47)(38,40,48,45);(1,4)(2,3)(5,9)(6,14)(7,12)(8,16)(10,13)(11,15)(17,20)(18,22)(19,28)(21,30)(23,25)(24,29)(26,31)(27,32)(33,37,39,36)(34,42,44,41)(35,43,46,47)(38,40,48,45);(1,5,7,2)(3,4,9,12)(6,13,15,8)(10,11,16,14)(17,21)(18,19)(20,30)(22,28)(23,24)(25,29)(26,27)(31,32)(33,35)(34,40)(36,47)(37,43)(38,41)(39,46)(42,48)(44,45);(1,6)(2,8)(3,10)(4,11)(5,13)(7,15)(9,16)(12,14)(17,22)(18,25)(19,29)(20,26)(21,28)(23,31)(24,32)(27,30)(33,38,39,48)(34,43,44,47)(35,41,46,42)(36,40,37,45)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8495591);