#8202018

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

4220 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)(3,11)(4,13)(5,10)(6,8)(7,14)(9,15)(12,16)(17,18,23,24)(19,25,20,29)(21,26,32,22)(27,28,30,31)(33,34)(35,36)(37,45)(38,40)(39,47)(41,43)(42,48)(44,46);(1,3,6,14)(2,7,8,11)(4,10,15,12)(5,13,16,9)(18,24)(21,32)(25,29)(27,30)(33,35,38,43)(34,36,40,41)(37,46,47,42)(39,48,45,44);(2,8)(5,16)(7,11)(9,13)(17,19,23,20)(18,25,24,29)(21,27,32,30)(22,31,26,28)(33,34,38,40)(35,41,43,36)(37,39,47,45)(42,48,46,44);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,7,16,11)(17,20,23,19)(18,25,24,29)(21,30,32,27)(22,31,26,28)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,46)(47,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,16)(10,15)(17,21)(18,26)(19,27)(20,30)(22,24)(23,32)(25,28)(29,31)(33,37)(34,39)(35,42)(36,44)(38,47)(40,45)(41,48)(43,46);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,7,15,11)(17,22,23,26)(18,21,24,32)(19,28,20,31)(25,30,29,27)(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,d^2=g,f^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, 8109214);