#456867

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
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×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1169 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,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, 364063);