#409984

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 ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1617 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)(12,25)(13,19)(14,18)(15,20)(16,23)(17,27)(22,32)(24,34)(26,35)(28,31)(29,33)(30,37)(36,39)(38,40);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,22)(13,28)(14,23)(15,24)(17,32)(18,34)(19,21)(20,31)(25,40)(26,37)(27,39)(29,38)(30,33)(35,36);(9,12)(10,17,20,33)(11,22)(13,29)(14,27)(15,26)(16,30,31,32)(18,35,19,25)(21,36,34,40)(23,39)(24,37)(28,38);(2,6)(5,8)(12,26)(17,33)(22,37)(25,35)(27,29)(30,32)(36,40)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257064, #72257104, #72257107
maximal normal subgroup of	#72257064, #72257104, #72257107
maximal quotient of	#72257355, #72264067

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317180);