#453025

order	512
nilpotency class	3
isoclinism	isoclinic class of #424610
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16076
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

753 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)(3,15,18,9)(4,19,22,29)(5,11,17,25)(6,13)(7,14)(8,12)(10,21,28,23)(16,27)(20,24)(26,31)(30,32);(1,3,7,17,8,18,6,5)(2,9,14,11,12,15,13,25)(4,20,21,31,22,32,23,16)(10,26,19,30,28,27,29,24);(1,4)(2,10,12,28)(3,16)(5,20)(6,21)(7,23)(8,22)(9,24,15,30)(11,27,25,26)(13,19,14,29)(17,32)(18,31);(1,5,6,18,8,17,7,3)(2,11,14,15,12,25,13,9)(4,20,21,31,22,32,23,16)(10,27,29,30,28,26,19,24)

Polycyclic presentation:

2^9|a^2=efgh,b^2=g,c^2=e,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,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, 360221);