#7489137

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45423	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1490 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=f,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=efi,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7396333);