#495622

order	512
nilpotency class	4
isoclinism	isoclinic class of #480476
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

975 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)(3,6)(4,5)(7,8)(9,10,15,17,14,18,12,25)(11,20,22,38,24,37,28,36)(13,19,21,31,27,35,29,34)(16,30,32,26,33,39,40,23);(9,11,21,30,14,24,29,39)(10,16,31,38,18,33,34,36)(12,22,27,23,15,28,13,26)(17,32,19,20,25,40,35,37);(2,5)(6,8)(9,12,14,15)(10,17,18,25)(11,22,24,28)(13,21,27,29)(16,32,33,40)(19,34,35,31)(20,36,37,38)(23,39,26,30);(1,3,4,7)(2,6,5,8)(10,18)(16,33)(17,25)(19,35)(20,37)(31,34)(32,40)(36,38)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402818);