#7206518

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,20,15)(12,16,13,22)(19,24,21,23)(25,26)(27,31)(28,38)(29,32)(30,35)(33,39)(34,37)(36,40);(1,3)(2,5)(4,7)(6,8)(9,11,14,20)(10,15,17,18)(12,21,13,19)(16,24,22,23)(25,27,30,36)(26,31,35,40)(28,37,39,29)(32,38,34,33);(3,7)(5,8)(26,32)(27,36)(29,37)(31,33)(34,35)(38,40);(2,6)(3,7)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28)(26,33)(27,37)(29,36)(30,39)(31,32)(34,40)(35,38);(3,7)(5,8)(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(27,36)(28,39)(31,40)(33,38)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=f,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=dfhi,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7113714);