#8201994

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
prime radical	#40758	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4223 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:

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

Polycyclic presentation:

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

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, 8109190);