#7569924

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#36060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1917 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,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, 7477120);