#432441

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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

959 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=f,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,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, 339637);