#450426

order	512
nilpotency class	3
isoclinism	isoclinic class of #417669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15967
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

925 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ef,c^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=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, 357622);