#465454

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14)(15,25)(16,30)(17,23)(18,24)(19,22)(20,28)(21,33)(26,35)(27,34)(29,36)(31,38)(32,37)(39,41)(40,44)(42,43);(13,15)(14,20,24,36)(16,27)(17,28)(18,26)(19,29)(21,34,38,44)(22,25,23,35)(30,43,37,39)(31,42)(32,40)(33,41);(5,7,8,12)(6,9,10,11)(15,26)(16,31)(17,19)(21,37)(22,23)(25,35)(27,41)(30,38)(32,33)(34,39)(40,42)(43,44);(2,4)(13,16,17,31,18,32,19,33)(14,21,22,37,24,38,23,30)(15,27,28,42,26,40,29,41)(20,34,25,39,36,44,35,43)

Polycyclic presentation:

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

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, 372650);