#7902829

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|c^2=gh,d^2=ghi,e^2=i,f^2=gi,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, 7810025);