#7524278

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514060
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	#36442	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=fi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7431474);