#8806744

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

4060 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=hi,e^2=hi,f^2=h,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, 8713940);