#8691816

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
prime radical	#59528	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4134 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8599012);