#8493057

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

4120 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)(4,8)(5,14)(6,11)(7,12)(10,15)(13,16)(17,18,23,27)(19,24,28,32)(20,22,26,31)(21,30,29,25)(33,34)(35,43)(36,37)(38,41)(39,42)(40,44)(45,48)(46,47);(1,3,7,4)(2,8,12,9)(5,13,15,11)(6,10,16,14)(19,28)(21,29)(24,32)(25,30)(35,44)(36,46)(37,47)(40,43);(1,3,7,4)(2,9,12,8)(5,11,15,13)(6,10,16,14)(17,19,23,28)(18,24,27,32)(20,29,26,21)(22,25,31,30)(33,35,39,44)(34,40,42,43)(36,38,46,48)(37,45,47,41);(1,4,7,3)(2,9,12,8)(5,13,15,11)(6,10,16,14)(17,20)(18,22)(19,29)(21,28)(23,26)(24,25)(27,31)(30,32)(33,36)(34,37)(35,38)(39,46)(40,45)(41,43)(42,47)(44,48);(1,5,7,15)(2,10,12,14)(3,11,4,13)(6,9,16,8)(17,21,23,29)(18,25,27,30)(19,20,28,26)(22,24,31,32)(33,37)(34,36)(35,45)(38,40)(39,47)(41,44)(42,46)(43,48);(1,6,7,16)(2,11,12,13)(3,10,4,14)(5,9,15,8)(17,22)(18,26)(19,25)(20,27)(21,24)(23,31)(28,30)(29,32)(33,38)(34,41)(35,46)(36,44)(37,40)(39,48)(42,45)(43,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8400253);