#458308

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
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

811 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365504);