#6373199

order	512
nilpotency class	3
isoclinism	isoclinic class of #60367
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2030 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)(3,8)(4,6)(5,7)(9,10,14,19,15,12,24,18)(11,16,23,30,22,20,34,29)(13,25,26,32,27,17,31,33)(21,35,37,39,36,28,38,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,28)(18,29)(19,30)(24,34)(25,35)(26,37)(27,36)(31,38)(32,39)(33,40);(9,12)(10,15)(11,20)(13,25)(14,18)(16,22)(17,27)(19,24)(21,35)(23,29)(26,32)(28,36)(30,34)(31,33)(37,39)(38,40);(1,4,5,3)(2,6,7,8)(9,13,14,26,15,27,24,31)(10,17,19,33,12,25,18,32)(11,21,23,37,22,36,34,38)(16,28,30,40,20,35,29,39);(11,22)(16,20)(21,36)(23,34)(28,35)(29,30)(37,38)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265151, #72308699, #72308721
maximal normal subgroup of	#72265151, #72308699, #72308721

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280395);