#433410

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

1016 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=ei,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^c=fi,g^c=gi,h^d=hi

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, 340606);