#8602780

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	#36794	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4275 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)(19,24)(20,21)(22,27)(23,25)(26,32)(28,29)(30,31)(34,37)(35,36)(41,45)(42,44);(1,2)(3,9)(4,7)(5,8)(6,10)(11,15)(12,13)(14,16)(17,19)(18,24)(20,30)(21,31)(22,25)(23,27)(26,28)(29,32)(33,34)(35,40)(36,43)(37,39)(38,41)(42,47)(44,48)(45,46);(3,10)(6,9)(11,16)(14,15)(18,25)(19,27)(21,32)(28,30)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(1,2)(3,6)(4,12)(5,8)(7,13)(9,10)(11,15)(14,16)(17,20,23,26)(18,21,25,32)(19,28,27,30)(22,31,24,29)(33,36,39,35)(34,40,37,43)(38,42,46,44)(41,48,45,47);(1,3,5,10)(2,6,8,9)(4,11,13,16)(7,14,12,15)(17,21)(18,26)(19,29)(20,25)(22,28)(23,32)(24,30)(27,31)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,4)(2,7)(3,11)(5,13)(6,14)(8,12)(9,15)(10,16)(17,22)(18,27)(19,25)(20,29)(21,28)(23,24)(26,31)(30,32)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48)

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=h,e^2=g,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, 8509976);