#363109

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

979 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270305);