#7487538

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1475 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=efi,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, 7394734);