#7027418

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996671
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	#23869	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1751 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6934614);