#7541617

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1472 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)(4,5)(6,8)(9,10)(11,14)(12,16)(13,15)(18,22)(19,25)(24,30)(26,28);(2,5)(6,7)(9,11,13,16)(10,14,15,12)(19,26)(21,27)(23,31)(24,30)(25,28)(29,32);(9,12,13,14)(10,11,15,16)(17,18)(19,23)(20,22)(21,24)(25,29)(26,31)(27,30)(28,32);(10,15)(11,16)(17,19,20,25)(18,23,22,29)(21,28,27,26)(24,32,30,31);(1,3,4,8)(2,6,5,7)(18,22)(23,29)(24,30)(31,32)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,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, 7448813);