#502097

order	512
nilpotency class	4
isoclinism	isoclinic class of #502097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25076
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,h^c=hi,e^d=ei

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, 409293);