#414682

order	512
nilpotency class	3
isoclinism	isoclinic class of #393989
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#392
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1618 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257064, #72257095
maximal normal subgroup of	#72257064, #72257095
maximal quotient of	#72257383

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321878);