#453805

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2804
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

633 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=efh,c^2=f,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361001);