#7487646

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309140
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	#39386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7394842);