#425191

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
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

1041 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)(21,34)(22,30)(23,24)(27,41)(28,29)(31,32)(37,38)(39,42)(40,44);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(13,14,18,23,19,24,17,20)(15,26,25,34,29,36,28,21)(16,30,32,38,33,37,31,22)(27,35,41,44,42,43,39,40);(2,4)(5,8,9,6)(7,11,12,10)(13,15)(14,21)(16,27)(17,25)(18,28)(19,29)(20,26)(22,35)(23,36)(24,34)(30,40)(31,41)(32,39)(33,42)(37,44)(38,43);(7,12)(10,11)(13,16,19,33)(14,22,24,38)(15,27,29,42)(17,32,18,31)(20,30,23,37)(21,35,34,43)(25,39,28,41)(26,40,36,44)

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,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=dei,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 332387);