#8131209

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	#57972	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=gi,e^2=gi,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^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, 8038405);