#453883

order	512
nilpotency class	3
isoclinism	isoclinic class of #427841
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2778
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16196
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

569 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=efh,c^2=g,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,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, 361079);