#8818535

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

4231 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=ghi,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, 8725731);