#7763445

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	#37546	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=i,e^2=hi,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7670641);