#501992

order	512
nilpotency class	4
isoclinism	isoclinic class of #501992
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25012
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28246	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

517 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)(3,13)(4,19)(5,22)(6,23)(7,11)(8,12)(9,16)(10,20)(14,15)(17,31)(18,28)(21,25)(24,32)(26,29)(27,30);(1,3)(2,9,12,29)(4,15)(5,16)(6,24)(7,26)(8,17)(10,28,23,14)(11,13,22,31)(18,21)(19,30,25,32)(20,27);(1,2)(3,14)(4,10)(5,11)(6,25)(7,22)(8,12)(9,27)(13,18)(15,31)(16,32)(17,28)(19,20)(21,23)(24,29)(26,30);(1,4,5,20,8,21,7,6)(2,10,11,19,12,23,22,25)(3,15,16,27,17,18,26,24)(9,28,13,30,29,14,31,32)

Polycyclic presentation:

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

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, 409188);