#8752261

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

4166 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)(7,10)(11,15)(14,16)(17,18,22,24)(19,25,20,23)(21,29,30,26)(27,31,28,32)(33,34)(35,39)(36,42)(37,40)(38,46)(41,44)(43,45)(47,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,19,22,20)(18,23,24,25)(21,28,30,27)(26,32,29,31)(33,35)(34,41)(36,43)(37,39)(38,47)(40,44)(42,48)(45,46);(2,6)(7,10)(8,12)(14,16)(18,24)(23,25)(26,29)(31,32)(33,36,40,38)(34,42,37,46)(35,43,44,47)(39,45,41,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,31)(28,32)(29,30)(33,37,40,34)(35,39,44,41)(36,42,38,46)(43,48,47,45);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,20,22,19)(18,25,24,23)(21,28,30,27)(26,32,29,31)(33,38,40,36)(34,42,37,46)(35,43,44,47)(39,48,41,45);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)(33,39)(34,35)(36,45)(37,44)(38,48)(40,41)(42,43)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=g,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, 8659457);