#427595

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	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1343 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei,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, 334791);