#407854

order	512
nilpotency class	3
isoclinism	isoclinic class of #396100
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2414
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14974
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

984 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315050);