#425185

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

1031 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,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, 332381);