#7877856

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	#59186	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4383 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)(11,19)(12,23)(13,18)(14,16)(15,22)(17,21)(20,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,39)(37,40);(1,2,5,4)(3,6,8,7)(10,16)(15,19)(17,23)(18,24)(25,26,29,32)(27,30,35,33)(28,31,37,40)(34,38,36,39);(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,39)(37,40);(1,4,5,2)(3,7,8,6)(9,12,14,21)(10,17,16,23)(11,20,22,13)(15,24,19,18)(25,27)(26,30)(28,36)(29,35)(31,39)(32,33)(34,37)(38,40);(9,13,14,20)(10,18,16,24)(11,21,22,12)(15,23,19,17)(25,28,29,37)(26,31,32,40)(27,34,35,36)(30,38,33,39)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7785052);