#9183880

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	#51968	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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)(3,8)(4,9)(5,13)(6,7)(10,15)(11,12)(14,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,34,40,44)(35,45,37,41)(36,42,46,47)(38,39,43,48);(2,7)(4,8)(10,13)(12,14)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,35,40,37)(34,41,44,45)(36,38,46,43)(39,42,48,47);(2,7)(3,9)(10,13)(11,16)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31)(33,36,40,46)(34,42,44,47)(35,38,37,43)(39,41,48,45);(1,3)(2,8)(4,7)(5,11)(6,9)(10,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,37,40,35)(34,41,44,45)(36,38,46,43)(39,47,48,42);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32)(33,38)(34,39)(35,46)(36,37)(40,43)(41,42)(44,48)(45,47);(1,5,6,15)(2,10,7,13)(3,11,9,16)(4,12,8,14)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)(33,39)(34,43)(35,42)(36,41)(37,47)(38,44)(40,48)(45,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9091076);