#7487875

order	512
nilpotency class	3
isoclinism	isoclinic class of #7487874
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#52004
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=i,d^2=fh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dfh,e^b=eh,f^b=fi,g^b=gi,d^c=dh,g^c=gi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7395071);