#7622409

order	512
nilpotency class	3
isoclinism	isoclinic class of #7535047
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3549
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	#60045
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1338 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)(3,12)(4,17)(5,9)(6,11)(7,14)(8,20)(10,21)(13,22)(15,24)(16,23)(18,28)(19,26)(25,30)(27,31)(29,32);(1,3,5,14,6,16,21,15)(2,7,10,12,11,24,9,23)(4,18,20,13,19,30,31,29)(8,25,17,22,27,28,26,32);(1,4,6,19)(2,8)(3,13)(5,20,21,31)(7,22,24,32)(9,26)(10,17)(11,27)(12,28,23,25)(14,30)(15,18)(16,29);(2,9,11,10)(7,12,24,23)(8,26,27,17)(22,28,32,25);(2,9,11,10)(4,19)(7,23,24,12)(8,17,27,26)(13,29)(18,30)(20,31)(22,28,32,25)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529605);