#8806708

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

4200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,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, 8713904);