#8479564

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

4041 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)(3,12)(4,10)(5,9)(6,8)(7,11)(13,15)(14,16)(17,18,22,23)(19,24,21,26)(20,28,29,25)(27,32,30,31)(36,44)(39,45)(43,47)(46,48);(1,3,7,6)(2,8,11,12)(4,13,5,14)(9,15,10,16)(18,23)(24,26)(25,28)(31,32)(33,34)(35,42)(36,45)(37,40)(38,41)(39,44)(43,46)(47,48);(1,4,7,5)(2,9,11,10)(3,13,6,14)(8,15,12,16)(17,19,22,21)(18,24,23,26)(20,27,29,30)(25,31,28,32)(33,35,38,40)(34,37,41,42)(36,43,44,47)(39,46,45,48);(1,5,7,4)(2,10,11,9)(3,13,6,14)(8,15,12,16)(17,20)(18,25)(19,27)(21,30)(22,29)(23,28)(24,31)(26,32)(33,34)(35,37)(36,39)(38,41)(40,42)(43,46)(44,45)(47,48);(1,2)(3,12)(4,9)(5,10)(6,8)(7,11)(13,16)(14,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32)(33,36)(34,39)(35,43)(37,46)(38,44)(40,47)(41,45)(42,48);(1,6,7,3)(2,8,11,12)(4,14,5,13)(9,15,10,16)(17,21,22,19)(18,26,23,24)(20,27,29,30)(25,31,28,32)(33,37)(34,40)(35,41)(36,46)(38,42)(39,47)(43,45)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386760);