#380179

order	512
nilpotency class	3
isoclinism	isoclinic class of #29751
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,d^c=de,f^c=fi,h^c=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, 287375);