#454658

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1792 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)(5,6,8,9)(7,10,11,12)(13,14,19,35,17,22,18,23)(15,24,29,25,27,20,28,37)(16,30,34,21,32,38,33,31)(26,40,44,39,42,41,43,36);(6,9)(10,12)(13,15)(14,20)(16,26)(17,27)(18,28)(19,29)(21,36)(22,24)(23,25)(30,41)(31,39)(32,42)(33,43)(34,44)(35,37)(38,40);(2,4)(5,7)(6,10)(8,11)(9,12)(13,14,17,22)(15,25,27,37)(16,31,32,21)(18,35,19,23)(20,29,24,28)(26,41,42,40)(30,34,38,33)(36,44,39,43);(7,11)(10,12)(13,16,17,32)(14,21,22,31)(15,26,27,42)(18,34,19,33)(20,36,24,39)(23,38,35,30)(25,40,37,41)(28,44,29,43)

Polycyclic presentation:

2^9|a^2=ef,c^2=i,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,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, 361854);