#380718

order	512
nilpotency class	3
isoclinism	isoclinic class of #29748
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1208 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)(13,14,19,24,17,22,18,23)(15,25,29,36,27,20,28,37)(16,30,34,39,32,21,33,40)(26,35,43,44,42,41,31,38);(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,15,19,29,17,27,18,28)(14,20,24,37,22,25,23,36)(16,31,34,26,32,43,33,42)(21,38,40,35,30,44,39,41);(2,4)(7,11)(10,12)(13,16)(14,21)(15,26)(17,32)(18,34)(19,33)(20,35)(22,30)(23,40)(24,39)(25,41)(27,42)(28,43)(29,31)(36,44)(37,38);(5,7,8,11)(6,10,9,12)(16,32)(21,30)(26,42)(31,43)(33,34)(35,41)(38,44)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^c=de,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, 287914);