#495648

order	512
nilpotency class	4
isoclinism	isoclinic class of #480482
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	C₂×#540	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

956 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,d^c=di,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, 402844);