#8505635

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

4187 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,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, 8412831);