#426226

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
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

957 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)(10,11)(13,14,19,23)(15,24,27,40)(16,28,33,43)(17,34,26,20)(18,35,30,22)(21,37,39,32)(25,41,42,36)(29,38,31,44);(1,2)(3,4)(5,6,8,9)(7,11,12,10)(13,15,19,27)(14,20,23,34)(16,29,33,31)(17,30,26,18)(21,38,39,44)(22,40,35,24)(25,37,42,32)(28,36,43,41);(2,4)(13,16,30,37,19,33,18,32)(14,21,35,43,23,39,22,28)(15,25,17,31,27,42,26,29)(20,36,40,44,34,41,24,38);(5,7,8,12)(6,10,9,11)(13,17,18,15,19,26,30,27)(14,20,22,24,23,34,35,40)(16,31,32,25,33,29,37,42)(21,36,28,38,39,41,43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=fi,d^2=e,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^b=ei,f^b=fi,g^b=gi,h^b=hi,h^d=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, 333422);