#8210027

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	#54952	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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)(3,11)(4,13)(5,10)(6,8)(7,14)(9,15)(12,16)(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32)(33,34)(35,44)(36,42)(37,38)(39,41)(40,43)(45,46)(47,48);(1,3,6,14)(2,7,8,11)(4,10,15,12)(5,13,16,9)(17,19)(18,23)(20,26)(21,29)(22,25)(24,32)(27,30)(28,31)(33,35,40,39)(34,41,43,44)(36,45,38,48)(37,46,42,47);(2,8)(5,16)(7,11)(9,13)(19,25)(23,26)(27,29)(31,32)(33,36,40,38)(34,42,43,37)(35,45,39,48)(41,47,44,46);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,16)(10,15)(17,20,22,18)(19,26,25,23)(21,28,30,24)(27,32,29,31)(33,37)(34,36)(35,46)(38,43)(39,47)(40,42)(41,45)(44,48);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,7,16,11)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,40,36)(34,37,43,42)(35,45,39,48)(41,47,44,46);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,7,15,11)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,39,40,35)(34,41,43,44)(36,45,38,48)(37,47,42,46)

Polycyclic presentation:

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

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, 8117223);