#371412

order	512
nilpotency class	3
isoclinism	isoclinic class of #24412
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×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1061 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=efh,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^c=fi,f^d=fi,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, 278608);