#7552473

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514000
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	#59043	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1720 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)(9,10)(11,15)(12,13)(14,16);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,28)(21,23)(22,25)(24,27)(29,32)(30,31);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,28,31,32);(9,11,12,16)(10,14,13,15)(17,20,22,29)(18,24,25,31)(19,27,21,30)(23,28,26,32);(1,4,5,3)(2,7,8,6)(18,25)(23,26)(24,31)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7459669);