#8603518

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	#45462	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4145 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8510714);