#430628

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

1117 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)(15,25)(16,30)(17,23)(18,24)(19,36)(20,28)(21,32)(22,33)(26,37)(27,44)(29,39)(31,40)(34,38)(35,41)(42,43);(2,3)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,31)(17,28)(18,34)(19,29)(21,40)(22,42)(23,25)(24,39)(26,33)(27,32)(30,44)(35,43)(36,38)(37,41);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,26)(17,32)(18,35)(19,33)(20,37)(22,24)(23,30)(25,42)(27,29)(28,43)(31,34)(36,41)(38,44)(39,40);(1,3,4,2)(6,9)(10,12)(13,16)(14,22)(15,27)(17,32)(18,33)(19,35)(20,37)(21,24)(23,41)(25,42)(26,29)(28,31)(30,36)(34,43)(38,40)(39,44)

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=ei,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, 337824);