#8158956

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	#37469	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3922 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:

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

Polycyclic presentation:

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

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, 8066152);