#7292769

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234345
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1338 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)(3,7)(4,8)(5,6)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26)(27,35)(28,38)(29,33)(30,34)(31,32)(36,40)(37,39);(2,6)(3,4)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,27,30,31)(26,32,34,35)(28,33,39,36)(29,38,40,37);(1,2)(3,8)(4,7)(5,6)(25,28,27,36,30,39,31,33)(26,29,32,37,34,40,35,38);(1,3,5,4)(2,7,6,8)(11,18)(14,17)(19,24)(22,23)(25,29,30,40)(26,33,34,36)(27,37,31,38)(28,35,39,32);(1,4,5,3)(2,7,6,8)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(26,34)(29,40)(32,35)(37,38)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=gi,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7199965);