#8757201

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

4170 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664397);