#428110

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1177 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)(3,4)(6,9)(10,11)(13,14)(15,25)(16,30)(17,23)(18,22)(19,24)(20,27)(21,32)(26,37)(28,36)(29,35)(31,39)(33,38)(34,43)(40,42)(41,44);(5,6,8,9)(7,11,12,10)(13,15,17,27,19,29,18,28)(14,20,22,25,24,36,23,35)(16,26,31,41,33,43,32,42)(21,37,38,40,39,34,30,44);(13,16,19,33)(14,21,24,39)(15,26,29,43)(17,32,18,31)(20,34,36,37)(22,30,23,38)(25,40,35,44)(27,42,28,41);(2,4)(5,7,8,12)(6,10,9,11)(13,15)(14,20)(16,26)(17,28)(18,27)(19,29)(21,34)(22,35)(23,25)(24,36)(30,44)(31,42)(32,41)(33,43)(37,39)(38,40)

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 335306);