#425858

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
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

1014 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:

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

Polycyclic presentation:

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

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, 333054);