#10411832

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3953 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)(3,4)(5,8)(6,9)(7,10)(11,12)(13,14)(15,16)(17,18,23,20)(19,25,21,24)(22,28,30,26)(27,31,29,32)(34,39)(36,43)(41,45)(44,48);(1,3)(2,4)(5,11)(6,7)(8,12)(9,10)(13,16)(14,15)(17,18,23,20)(19,25,21,24)(22,26,30,28)(27,32,29,31)(33,34)(35,36)(37,41)(38,39)(40,43)(42,44)(45,46)(47,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,35,38,40)(34,36,39,43)(37,42,46,47)(41,44,45,48);(1,2)(3,4)(5,13)(6,9)(7,10)(8,14)(11,16)(12,15)(17,20,23,18)(19,25,21,24)(22,26,30,28)(27,31,29,32)(33,34)(35,36)(37,45)(38,39)(40,43)(41,46)(42,48)(44,47);(1,4)(2,7)(3,9)(5,12)(6,10)(8,15)(11,13)(14,16)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31)(33,36)(34,40)(35,39)(37,44)(38,43)(41,47)(42,45)(46,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,37,38,46)(34,41,39,45)(35,42,40,47)(36,44,43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10319028);