#7227645

order	512
nilpotency class	3
isoclinism	isoclinic class of #7199982
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 ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1337 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=fh,e^2=f,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=eh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7134841);