#453869

order	512
nilpotency class	3
isoclinism	isoclinic class of #425083
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2576
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16175
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

545 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)(3,4,16,9)(5,20)(6,13)(7,12)(8,14)(10,18,23,19)(11,22)(15,29,30,32)(17,26,25,31)(21,27)(24,28);(1,3,8,23,6,16,7,10)(2,9,12,19,13,4,14,18)(5,15,22,26,21,30,24,31)(11,17,20,32,28,25,27,29);(1,4,6,9)(2,10)(3,14)(5,17,21,25)(7,18,8,19)(11,26)(12,16)(13,23)(15,20)(22,29,24,32)(27,30)(28,31);(1,5,7,22,6,21,8,24)(2,11,12,27,13,28,14,20)(3,15,10,26,16,30,23,31)(4,17,18,29,9,25,19,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361065);