#405116

order	512
nilpotency class	3
isoclinism	isoclinic class of #394040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60359	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2303 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312725, #72313120, #72313127
maximal normal subgroup of	#72312725, #72313120, #72313127
maximal quotient of	#72317003, #72317042

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312312);