#406271

order	512
nilpotency class	3
isoclinism	isoclinic class of #394321
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60444	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1714 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)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,28)(22,37)(23,32)(25,36)(27,40)(29,35)(30,34)(31,39)(33,38);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,33)(18,21)(19,36)(20,32)(22,29)(26,31)(28,39)(30,38)(34,37)(35,40);(9,12,13,28,15,30,14,29)(10,17,19,26,20,35,18,34)(11,22,24,38,25,39,23,27)(16,31,21,33,32,37,36,40);(2,6)(5,8)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,30,29)(16,21,32,36)(17,34,35,26)(22,27,39,38)(31,33,37,40)

Polycyclic presentation:

2^9|b^2=h,c^2=f,d^2=i,e^2=i,f^2=i,g^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,g^b=gi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312750, #72313120, #72313127
maximal normal subgroup of	#72312750, #72313120, #72313127
maximal quotient of	#72316999, #72317040

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313467);