#7927641

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	#35393	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4397 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)(9,10,13,16)(11,14,19,23)(12,15,20,24)(17,21,18,22)(26,31)(27,33)(28,36)(38,40);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,23)(17,20)(21,24)(25,26)(27,34)(28,32)(29,33)(30,31)(35,40)(36,39)(37,38);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,30,33)(26,29,31,34)(28,37,36,35)(32,40,39,38);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,28,30,36)(26,32,31,39)(27,35,33,37)(29,38,34,40);(9,12)(10,15)(11,18)(13,20)(14,22)(16,24)(17,19)(21,23)(25,26)(27,29)(28,32)(30,31)(33,34)(35,38)(36,39)(37,40);(9,11)(10,14)(12,17)(13,19)(15,21)(16,23)(18,20)(22,24)(25,29)(26,33)(27,31)(28,38)(30,34)(32,37)(35,39)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7834837);