#8534936

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

4108 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8442132);