#8202228

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

4156 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=gh,e^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, 8109424);