#10181359

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	#40649	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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)(3,10)(4,13)(5,12)(6,8)(7,14)(9,15)(11,16)(17,18,23,20)(19,24,27,28)(21,25,31,30)(22,29,32,26);(1,3,6,14)(2,7,8,10)(4,12,15,11)(5,13,16,9)(17,19)(18,24)(20,28)(21,26)(22,25)(23,27)(29,31)(30,32)(33,34,39,37)(35,42,36,40)(38,41,46,45)(43,48,44,47);(2,8)(4,15)(7,10)(11,12)(18,20)(19,27)(25,30)(26,29)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,16)(11,14)(17,20,23,18)(19,28,27,24)(21,25,31,30)(22,29,32,26)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,3)(2,10)(4,11)(5,9)(6,14)(7,8)(12,15)(13,16)(17,21)(18,25)(19,29)(20,30)(22,28)(23,31)(24,32)(26,27)(33,37,39,34)(35,42,36,40)(38,41,46,45)(43,47,44,48);(1,5)(2,11)(3,13)(4,7)(6,16)(8,12)(9,14)(10,15)(17,22)(18,26)(19,25)(20,29)(21,24)(23,32)(27,30)(28,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=i,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10088555);