#7576407

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1623 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)(3,7)(4,6)(5,8)(10,14)(11,12)(17,18,21,24)(19,25,27,22)(20,23,28,31)(26,32,29,30);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16);(1,4,5,3)(2,6,8,7)(9,11,13,12)(10,16,14,15)(17,19,21,27)(18,22,24,25)(20,29,28,26)(23,32,31,30);(2,8)(6,7)(9,12,13,11)(10,15,14,16)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,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, 7483603);