#7234277

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207866
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×#496	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1282 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)(3,6)(4,7)(5,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(26,30)(27,33)(28,35)(31,36)(32,39)(37,38);(3,4)(6,7)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(26,31)(27,33)(30,36)(34,40);(1,3,5,4)(2,6,8,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,26,28,30,29,31,35,36)(27,32,34,37,33,38,40,39);(1,3,5,4)(2,7,8,6)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,27,29,33)(26,32,31,38)(28,34,35,40)(30,37,36,39);(1,4,5,3)(2,7,8,6)(10,18)(15,19)(16,17)(23,24)(27,33)(32,38)(34,40)(37,39)

Polycyclic presentation:

2^9|b^2=f,c^2=fghi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,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, 7141473);