#452792

order	512
nilpotency class	3
isoclinism	isoclinic class of #418488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

883 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359988);