#456805

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

1003 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)(8,9)(11,12)(13,14,19,25,18,24,20,26)(15,27,30,41,29,21,31,40)(16,32,35,39,34,43,36,22)(17,37,33,44,28,42,38,23);(1,2)(3,4)(6,9)(10,12)(13,15,19,30,18,29,20,31)(14,21,25,40,24,27,26,41)(16,33,35,28,34,38,36,17)(22,42,32,23,39,37,43,44);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,34)(14,22,24,39)(15,28,29,17)(19,36,20,35)(21,23,27,44)(25,43,26,32)(30,33,31,38)(37,40,42,41);(7,11)(10,12)(13,17,18,28)(14,23,24,44)(15,16,29,34)(19,33,20,38)(21,39,27,22)(25,37,26,42)(30,35,31,36)(32,40,43,41)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=e,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,f^c=fi,g^c=gi,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, 364001);