#8975189

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	#2615	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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:

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

Polycyclic presentation:

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

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, 8882385);