#451040

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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1362 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)(9,10,15,18,13,19,14,20)(11,16,24,31,22,32,23,30)(12,25,27,17,26,34,28,35)(21,36,38,29,37,40,39,33);(2,5)(3,7)(9,11,13,22)(10,16,19,32)(12,21,26,37)(14,24,15,23)(17,33,35,29)(18,30,20,31)(25,36,34,40)(27,39,28,38);(9,12)(10,17)(11,21)(13,26)(14,28)(15,27)(16,29)(18,34)(19,35)(20,25)(22,37)(23,39)(24,38)(30,36)(31,40)(32,33);(1,3,4,7)(2,6,5,8)(10,18,19,20)(11,22)(16,30,32,31)(17,34,35,25)(21,37)(23,24)(29,36,33,40)(38,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=ei,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	#72284199, #72284254, #72286126, …
maximal normal subgroup of	#72284199, #72284254, #72286126, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358236);