#470766

order	512
nilpotency class	3
isoclinism	isoclinic class of #460311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2500
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16699
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

752 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eh,c^2=ef,e^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,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=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, 377962);