#7125752

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1370 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=hi,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 7032948);