#501632

order	512
nilpotency class	4
isoclinism	isoclinic class of #501631
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24726
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

661 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=ei,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=gh,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, 408828);