#404260

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	#62855	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,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	#72282239

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311456);