#451023

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2404
prime radical	#34633	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1393 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)(3,6)(4,5)(7,8)(9,10,13,17,14,18,27,19)(11,20,23,30,22,15,26,29)(12,16,25,34,24,33,37,32)(21,28,36,40,35,31,38,39);(2,5)(3,7)(9,11)(10,15)(12,21)(13,26)(14,22)(16,31)(17,30)(18,20)(19,29)(23,27)(24,35)(25,38)(28,33)(32,39)(34,40)(36,37);(9,12)(10,16)(11,21)(13,25)(14,24)(15,28)(17,34)(18,33)(19,32)(20,31)(22,35)(23,36)(26,38)(27,37)(29,40)(30,39);(1,3,4,7)(2,6,5,8)(10,17,18,19)(12,24)(15,29,20,30)(16,32,33,34)(21,35)(25,37)(28,39,31,40)(36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318131, #72318305, #72318340
maximal normal subgroup of	#72318131, #72318305, #72318340

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358219);