#464641

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1528 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)(5,6)(7,10)(8,9)(11,12)(13,14,20,24,18,16,19,23)(15,25,30,26,28,21,29,37)(17,32,35,31,33,38,34,22)(27,41,44,36,42,40,43,39);(6,9)(10,12)(13,15,19,29,18,28,20,30)(14,21,23,26,16,25,24,37)(17,27,35,44,33,42,34,43)(22,36,32,40,31,39,38,41);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16,18,14)(15,26,28,37)(17,22,33,31)(19,23,20,24)(21,30,25,29)(27,41,42,40)(32,35,38,34)(36,44,39,43);(7,11)(10,12)(13,17)(14,22)(15,27)(16,31)(18,33)(19,35)(20,34)(21,36)(23,32)(24,38)(25,39)(26,40)(28,42)(29,44)(30,43)(37,41)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 371837);