#192559

order	512
nilpotency class	3
isoclinism	isoclinic class of #18192
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

881 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)(9,10,14,18,15,19,13,29)(11,20,22,16,24,31,23,32)(12,25,26,17,28,33,27,34)(21,36,38,35,39,30,37,40);(1,2,6,3)(4,7,5,8)(9,11,13,23,15,24,14,22)(10,16,29,20,19,32,18,31)(12,21,27,37,28,39,26,38)(17,30,25,35,34,36,33,40);(1,3,6,2)(4,8,5,7)(9,12,15,28)(10,17,19,34)(11,21,24,39)(13,27,14,26)(16,30,32,36)(18,33,29,25)(20,35,31,40)(22,38,23,37);(1,4,6,5)(2,8,3,7)(9,13,15,14)(10,18,19,29)(11,22,24,23)(12,26,28,27)(16,20,32,31)(17,25,34,33)(21,37,39,38)(30,40,36,35)

Polycyclic presentation:

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

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, 99755);