#474209

order	512
nilpotency class	3
isoclinism	isoclinic class of #457680
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2665
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16908
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

553 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=ef,d^2=e,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,f^b=fi,d^c=de,g^c=gi,h^c=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, 381405);