#8731082

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	#55750	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4029 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8638278);