#8157703

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	#40982	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,e^2=g,f^2=h,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, 8064899);