#8811215

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4039 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)(4,8)(9,13)(12,15)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(33,34,38,41)(35,36,39,44)(37,40,46,48)(42,43,45,47);(1,2,6,7)(3,8,11,4)(5,13,14,9)(10,12,16,15)(21,29)(25,28)(27,30)(31,32)(33,35,38,39)(34,36,41,44)(37,45,46,42)(40,47,48,43);(1,3,6,11)(2,8,7,4)(5,10,14,16)(9,15,13,12)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,35)(34,36)(37,42)(38,39)(40,43)(41,44)(45,46)(47,48);(1,4,6,8)(2,3,7,11)(5,12,14,15)(9,10,13,16)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,45)(42,48)(46,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,13)(14,16)(17,21,23,29)(18,25,20,28)(19,27,22,30)(24,31,26,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,45)(42,48)(46,47);(1,5)(2,9)(3,10)(4,12)(6,14)(7,13)(8,15)(11,16)(17,22,23,19)(18,26,20,24)(21,30,29,27)(25,32,28,31)(33,37,38,46)(34,40,41,48)(35,42,39,45)(36,43,44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=gh,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8718411);