#497807

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

443 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)(7,8)(9,10,15,19,16,23,31,38,17,22,12,30,14,21,13,20)(11,24,25,39,28,18,26,36,29,37,32,34,27,35,40,33);(1,2)(3,7)(4,8)(5,6)(9,11)(10,18,22,35)(12,25)(13,26)(14,27)(15,32)(16,28)(17,29)(19,33,30,36)(20,34,38,39)(21,24,23,37)(31,40);(1,3,5,4)(2,7,6,8)(9,12,14,31,17,15,16,13)(10,19,21,20,22,30,23,38)(11,25,27,40,29,32,28,26)(18,33,24,34,35,36,37,39);(1,4,5,3)(2,7,6,8)(9,13,14,12,17,31,16,15)(10,20,21,30,22,38,23,19)(11,26,27,25,29,40,28,32)(18,34,24,36,35,39,37,33)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=deh,g^b=gi,d^c=di,g^c=gi

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, 405003);