#449714

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

1536 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)(11,15)(12,22,25,16)(13,26)(14,17)(18,29)(19,30,37,40)(20,32)(21,35)(23,39,38,33)(24,34,28,36)(27,31);(1,2,4,5)(3,6,7,8)(9,11,14,20)(10,15,17,32)(12,23,25,38)(13,27,29,21)(16,33,22,39)(18,35,26,31)(19,28,37,24)(30,36,40,34);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,28)(14,25)(15,30)(17,22)(18,36)(20,37)(21,38)(23,27)(24,29)(26,34)(31,39)(32,40)(33,35);(2,5)(6,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,30)(20,32)(21,31)(22,25)(23,33)(24,34)(26,29)(27,35)(28,36)(37,40)(38,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=e,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, 356910);