#8505565

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

3958 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=i,e^2=gh,f^2=hi,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, 8412761);