#8602630

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	#54328	C₂
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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8509826);