#474722

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
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	#2782
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

596 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)(3,7)(4,5)(6,8)(9,10,14,30,15,20,32,19)(11,21,13,16,23,37,29,18)(12,24,26,35,27,17,31,38)(22,34,40,33,25,39,28,36);(2,5)(6,7)(9,11,14,13,15,23,32,29)(10,16,19,21,20,18,30,37)(12,25,31,40,27,22,26,28)(17,33,38,39,24,36,35,34);(1,3,4,8)(2,6,5,7)(9,12,15,27)(10,17,20,24)(11,22,23,25)(13,28,29,40)(14,31,32,26)(16,33,18,36)(19,38,30,35)(21,39,37,34);(3,8)(6,7)(9,13,14,23,15,29,32,11)(10,18,30,21,20,16,19,37)(12,25,26,28,27,22,31,40)(17,34,38,33,24,39,35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381918);