#192289

order	512
nilpotency class	3
isoclinism	isoclinic class of #18414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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×#385
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

960 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=h,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99485);