#192565

order	512
nilpotency class	3
isoclinism	isoclinic class of #18284
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

870 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99761);