#8135332

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	#48580	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3974 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)(3,9)(4,12)(5,10)(6,7)(8,13)(11,14)(15,16)(17,18,23,26)(19,24,20,21)(22,30,31,25)(27,29,28,32)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,44)(47,48);(2,7)(3,10)(8,12)(11,16)(17,19,23,20)(18,24,26,21)(22,27,31,28)(25,32,30,29)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,48)(46,47);(2,7)(3,10)(4,13)(14,15)(19,20)(21,24)(27,28)(29,32)(33,36,38,42)(34,39,41,35)(37,44,46,48)(40,47,45,43);(1,3)(2,5)(4,11)(6,10)(7,9)(8,14)(12,15)(13,16)(17,20,23,19)(18,21,26,24)(22,28,31,27)(25,29,30,32)(33,36)(34,35)(37,44)(38,42)(39,41)(40,43)(45,47)(46,48);(1,4,6,13)(2,8,7,12)(3,11,10,16)(5,14,9,15)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,30)(31,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,5)(2,3)(4,14)(6,9)(7,10)(8,11)(12,16)(13,15)(17,22,23,31)(18,25,26,30)(19,27,20,28)(21,29,24,32)(33,37,38,46)(34,40,41,45)(35,43,39,47)(36,44,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=i,e^2=h,f^2=ghi,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^c=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, 8042528);