#7231211

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1390 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)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26,31,32,30,35,27,36)(28,38,37,29,34,39,40,33);(2,6)(3,4)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,18)(22,23)(25,27,30,31)(26,32,35,36)(28,37,34,40)(29,38,33,39);(1,3,5,4)(2,7,6,8)(25,28,30,34)(26,33,35,29)(27,37,31,40)(32,39,36,38);(2,6)(7,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,29,30,33)(26,34,35,28)(27,38,31,39)(32,40,36,37);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(26,35)(29,33)(32,36)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7138407);