#8371258

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	#54235	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18,23,20)(19,24,26,22)(21,25,31,28)(27,32,29,30)(34,40)(36,41)(42,47)(44,48);(2,7)(3,11)(4,14)(13,16)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,30,28,32)(33,34,39,40)(35,41,37,36)(38,42,46,47)(43,48,45,44);(2,7)(5,9)(8,10)(13,16)(19,26)(22,24)(27,29)(30,32)(33,35,39,37)(34,41,40,36)(38,45,46,43)(42,44,47,48);(1,3,6,11)(2,8,7,10)(4,15,14,12)(5,13,9,16)(17,20,23,18)(19,22,26,24)(21,25,31,28)(27,32,29,30)(33,36)(34,35)(37,40)(38,44)(39,41)(42,43)(45,47)(46,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,16)(11,15)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,30,24,32)(33,37,39,35)(34,41,40,36)(38,45,46,43)(42,48,47,44);(1,5,6,9)(2,4,7,14)(3,13,11,16)(8,15,10,12)(17,22)(18,19)(20,26)(21,30)(23,24)(25,27)(28,29)(31,32)(33,38,39,46)(34,42,40,47)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=hi,e^2=ghi,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,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, 8278454);