#421565

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

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

913 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,20)(16,29)(17,35)(18,23)(19,24)(21,32)(22,27)(25,36)(26,43)(28,41)(30,39)(31,37)(33,42)(34,40)(38,44);(2,3)(5,7,8,11)(6,9,10,12)(13,15,18,17)(14,20,23,35)(16,30,32,44)(19,28,36,27)(21,38,29,39)(22,24,41,25)(26,34,31,33)(37,42,43,40);(1,2,4,3)(6,10)(9,12)(13,16)(14,21)(15,26)(17,31)(18,32)(19,34)(20,37)(22,39)(23,29)(24,42)(25,40)(27,44)(28,30)(33,36)(35,43)(38,41);(1,3,4,2)(13,17,18,15)(14,22,23,41)(16,31,32,26)(19,28,36,27)(20,24,35,25)(21,39,29,38)(30,33,44,34)(37,42,43,40)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=ei,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, 328761);