#8534629

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

4324 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,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, 8441825);