#380698

order	512
nilpotency class	3
isoclinism	isoclinic class of #29742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1220 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)(6,9)(10,12)(13,14)(15,20)(16,29,33,40)(17,23)(18,24)(19,22)(21,31,39,32)(25,41,42,38)(26,37)(27,35)(28,36)(30,34,43,44);(5,6,8,9)(7,10,11,12)(13,15,18,28,19,26,17,27)(14,20,24,36,22,37,23,35)(16,30,32,42,33,43,31,25)(21,38,40,44,39,41,29,34);(2,4)(7,11)(10,12)(13,16)(14,21)(15,25)(17,32)(18,31)(19,33)(20,34)(22,39)(23,40)(24,29)(26,42)(27,30)(28,43)(35,38)(36,41)(37,44);(5,7,8,11)(6,10,9,12)(14,22)(15,26)(21,39)(23,24)(25,42)(27,28)(29,40)(30,43)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287894);