#472582

order	512
nilpotency class	3
isoclinism	isoclinic class of #469927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2605
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16826
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

667 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)(3,13,15,8)(4,9)(5,12)(6,10)(7,11)(14,22,30,29)(16,23,17,24)(18,32,31,25)(19,26)(20,27)(21,28);(1,3)(2,8)(4,18,20,31)(5,16)(6,17)(7,15)(9,25,27,32)(10,23)(11,13)(12,24)(14,19,30,21)(22,28,29,26);(1,4)(2,9)(3,14,15,30)(5,19)(6,21)(7,20)(8,22,13,29)(10,28)(11,27)(12,26)(16,18,17,31)(23,25,24,32);(2,10,11,12)(8,23,13,24)(9,26,27,28)(22,32,29,25)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=f,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,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, 379778);