#8841359

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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:

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

Polycyclic presentation:

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

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, 8748555);