#403948

order	512
nilpotency class	3
isoclinism	isoclinic class of #394046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2677 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)(11,23)(12,27)(13,31)(14,21)(15,20)(16,22)(17,24)(18,29)(19,32)(25,37)(26,36)(28,39)(30,38)(33,35)(34,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,13)(14,25)(15,24)(16,26)(18,35)(19,27)(20,36)(21,23)(22,37)(28,34)(29,32)(30,33)(31,39)(38,40);(2,5)(6,8)(9,12)(10,18)(11,13)(14,28)(15,29)(16,30)(17,35)(19,23)(20,38)(21,27)(22,39)(24,32)(25,34)(26,33)(31,37)(36,40);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,12)(14,32)(15,34)(16,33)(17,27)(18,23)(20,31)(21,35)(22,40)(24,28)(25,29)(26,30)(36,39)(37,38)

Polycyclic presentation:

2^9|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,h^a=hi,c^b=cg,e^b=ei,f^c=fi,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	#72282223

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311144);