#7206489

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200002
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	#36102	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1709 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi

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, 7113685);