#449782

order	512
nilpotency class	3
isoclinism	isoclinic class of #417558
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×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1420 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=fi,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, 356978);