#8295056

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

4087 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8202252);