#7917802

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	#47866	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4481 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=ghi,e^2=g,f^2=ghi,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, 7824998);