#7622405

order	512
nilpotency class	3
isoclinism	isoclinic class of #7535046
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	#59945
prime radical	#24288	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1156 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)(3,13)(4,18)(5,11)(6,9)(7,12)(8,16)(10,20)(14,24)(15,29)(17,22)(19,25)(21,26)(23,30)(27,31)(28,32);(1,3,5,16,7,17,6,14)(2,8,9,13,12,24,11,22)(4,15,20,31,19,30,21,28)(10,23,18,27,26,29,25,32);(2,9,12,11)(3,14,17,16)(10,18,26,25)(15,28,30,31);(2,9,12,11)(4,19)(8,22,24,13)(10,25,26,18)(15,30)(20,21)(23,27,29,32)(28,31);(1,4,5,20,7,19,6,21)(2,10,9,18,12,26,11,25)(3,15,16,31,17,30,14,28)(8,23,13,27,24,29,22,32)

Polycyclic presentation:

2^9|b^2=f,c^2=fgh,d^2=h,e^2=f,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^d=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, 7529601);