#451039

order	512
nilpotency class	3
isoclinism	isoclinic class of #417566
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1306 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)(9,10,14,18,15,19,13,20)(11,21,24,32,25,16,23,31)(12,26,27,17,28,34,29,35)(22,36,37,33,38,40,39,30);(2,5)(3,7)(9,11,15,25)(10,16,19,21)(12,22,28,38)(13,24,14,23)(17,33,35,30)(18,32,20,31)(26,40,34,36)(27,39,29,37);(9,12)(10,17)(11,22)(13,29)(14,27)(15,28)(16,30)(18,34)(19,35)(20,26)(21,33)(23,39)(24,37)(25,38)(31,36)(32,40);(1,3,4,7)(2,6,5,8)(10,18,19,20)(16,31,21,32)(17,34,35,26)(30,36,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284166, #72284254, #72317957, …
maximal normal subgroup of	#72284166, #72284254, #72317957, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358235);