#449713

order	512
nilpotency class	3
isoclinism	isoclinic class of #417534
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
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

1450 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:

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

Polycyclic presentation:

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

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, 356909);