#404302

order	512
nilpotency class	3
isoclinism	isoclinic class of #394046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62856	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2375 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,h^b=hi,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282010
maximal normal subgroup of	#72282010
maximal quotient of	#72282084

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311498);