#8649075

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	#39308	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=gi,e^2=g,f^2=hi,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, 8556271);