#8202014

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

4102 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)(3,10)(4,7)(5,12)(6,8)(9,14)(11,15)(13,16)(17,18,24,27)(19,28,23,25)(20,22,21,26)(29,31,30,32)(33,34)(35,36)(37,45)(38,40)(39,47)(41,43)(42,48)(44,46);(1,3,7,6)(2,8,4,10)(5,13,14,11)(9,16,12,15)(17,19,24,23)(18,25,27,28)(20,30,21,29)(22,31,26,32)(33,35,38,43)(34,36,40,41)(37,46,47,42)(39,48,45,44);(1,2,7,4)(3,8,6,10)(5,9,14,12)(11,15,13,16)(17,20,24,21)(18,26,27,22)(19,29,23,30)(25,31,28,32)(33,34,38,40)(35,41,43,36)(37,39,47,45)(42,48,46,44);(1,4,7,2)(3,8,6,10)(5,12,14,9)(11,15,13,16)(17,21,24,20)(18,22,27,26)(19,30,23,29)(25,32,28,31)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,46)(47,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,22)(18,20)(19,31)(21,27)(23,32)(24,26)(25,30)(28,29)(33,37)(34,39)(35,42)(36,44)(38,47)(40,45)(41,48)(43,46);(1,6,7,3)(2,8,4,10)(5,13,14,11)(9,15,12,16)(17,23,24,19)(18,25,27,28)(20,30,21,29)(22,32,26,31)(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=h,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, 8109210);