#456035

order	512
nilpotency class	3
isoclinism	isoclinic class of #454109
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2670
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16273
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

901 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:

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

Polycyclic presentation:

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

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, 363231);