#7569892

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	#35927	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1883 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)(11,12)(14,16)(17,18,21,25)(19,26,29,23)(20,30,22,24)(27,28,32,31);(2,5)(6,7)(9,10,13,15)(11,16,12,14)(17,18)(19,27)(20,24)(21,25)(22,30)(23,28)(26,31)(29,32);(9,11,13,12)(10,14,15,16)(17,18)(19,23)(20,30)(21,25)(22,24)(26,29)(27,28)(31,32);(9,12,13,11)(10,14,15,16)(17,19,21,29)(18,23,25,26)(20,28,22,31)(24,32,30,27);(1,3,4,8)(2,6,5,7)(17,20,21,22)(18,24,25,30)(19,28,29,31)(23,32,26,27)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=g,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, 7477088);