#185894

order	512
nilpotency class	3
isoclinism	isoclinic class of #18192
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62788	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2237 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,d^c=dh,e^c=eh,g^c=gh

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, 93090);