#6372160

order	512
nilpotency class	3
isoclinism	isoclinic class of #60460
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	C₂×#575	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2002 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)(9,10,16,21,14,20,15,33)(11,22,27,39,25,17,26,35)(12,28,24,37,29,40,30,18)(13,19,32,36,23,38,31,34);(2,5)(6,8)(9,11,16,27,14,25,15,26)(10,17,21,35,20,22,33,39)(12,13,24,32,29,23,30,31)(18,36,28,38,37,34,40,19);(9,12,15,30,14,29,16,24)(10,18,21,28,20,37,33,40)(11,23,26,32,25,13,27,31)(17,34,35,19,22,36,39,38);(1,3,4,7)(2,6,5,8)(9,13,14,23)(10,19,20,38)(11,24,25,30)(12,27,29,26)(15,31,16,32)(17,18,22,37)(21,36,33,34)(28,39,40,35);(12,29)(13,23)(18,37)(19,38)(24,30)(28,40)(31,32)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239527, #72239792, #72308712
maximal normal subgroup of	#72239527, #72239792, #72308712

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279356);