#8588392

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3965 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)(19,26)(20,28)(21,22)(23,24)(25,31)(27,30)(29,32)(33,34,39,42)(35,40,43,37)(36,38,41,47)(44,45,48,46);(1,3,7,9)(2,4,5,12)(6,10,15,16)(8,11,13,14)(17,19,23,22)(18,21,24,26)(20,29,31,27)(25,32,28,30)(33,35)(34,40)(36,44)(37,42)(38,45)(39,43)(41,48)(46,47);(2,5)(3,9)(8,13)(10,16)(18,24)(21,26)(25,28)(30,32)(35,43)(37,40)(44,48)(45,46);(1,4)(2,3)(5,9)(6,14)(7,12)(8,16)(10,13)(11,15)(17,20,23,31)(18,25,24,28)(19,27,22,29)(21,30,26,32)(33,36)(34,38)(35,44)(37,46)(39,41)(40,45)(42,47)(43,48);(1,5,7,2)(3,4,9,12)(6,13,15,8)(10,11,16,14)(17,21)(18,19)(20,30)(22,24)(23,26)(25,27)(28,29)(31,32)(33,37)(34,35)(36,46)(38,44)(39,40)(41,45)(42,43)(47,48);(1,6)(2,8)(3,10)(4,11)(5,13)(7,15)(9,16)(12,14)(17,22,23,19)(18,21,24,26)(20,27,31,29)(25,32,28,30)(33,38)(34,41)(35,45)(36,42)(37,44)(39,47)(40,48)(43,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,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, 8495588);