#363108

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1023 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)(11,18,23,36)(12,24,28,17)(13,29,30,16)(14,19)(15,20)(21,35)(22,38)(25,33)(26,31)(27,39,37,34)(32,40);(1,2,4,6)(3,7,8,5)(9,11,21,30,15,23,14,13)(10,16,19,18,20,29,35,36)(12,25,37,26,28,22,27,40)(17,32,34,38,24,31,39,33);(1,3,4,8)(2,5,6,7)(9,12,15,28)(10,17,20,24)(11,22,23,25)(13,26,30,40)(14,27,21,37)(16,31,29,32)(18,33,36,38)(19,34,35,39);(9,13,14,23,15,30,21,11)(10,18,35,16,20,36,19,29)(12,26,27,25,28,40,37,22)(17,33,39,31,24,38,34,32)

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=g,d^2=ef,e^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, 270304);