#6371840

order	512
nilpotency class	3
isoclinism	isoclinic class of #60372
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2179 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,g^c=gi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279036);