#7355391

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324532
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	#45593	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1495 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=fgi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7262587);