#8621783

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	#53559	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4086 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)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(17,18,23,26)(19,21,24,29)(20,22,25,31)(27,28,30,32)(33,34)(35,41)(36,46)(37,38)(39,42)(40,43)(44,48)(45,47);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(17,19,23,24)(18,21,26,29)(20,27,25,30)(22,28,31,32)(33,35,40,45)(34,41,43,47)(36,42,37,44)(38,48,46,39);(2,5)(8,10)(9,12)(15,16)(19,24)(21,29)(27,30)(28,32)(33,36,40,37)(34,38,43,46)(35,44,45,42)(39,47,48,41);(1,4,7,14)(2,9,5,12)(3,11,6,13)(8,15,10,16)(17,20)(18,22)(19,27)(21,28)(23,25)(24,30)(26,31)(29,32)(33,37,40,36)(34,38,43,46)(35,42,45,44)(39,47,48,41);(1,5,7,2)(3,8,6,10)(4,12,14,9)(11,15,13,16)(17,21,23,29)(18,24,26,19)(20,28,25,32)(22,30,31,27)(33,38)(34,36)(35,39)(37,43)(40,46)(41,44)(42,47)(45,48);(1,6,7,3)(2,8,5,10)(4,11,14,13)(9,16,12,15)(17,22)(18,25)(19,28)(20,26)(21,30)(23,31)(24,32)(27,29)(33,39)(34,42)(35,38)(36,47)(37,41)(40,48)(43,44)(45,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=h,e^2=hi,f^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, 8528979);