#406231

order	512
nilpotency class	3
isoclinism	isoclinic class of #394283
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1701 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)(3,5)(4,6)(7,8)(9,10)(11,20)(12,25)(13,29)(14,18)(15,19)(16,22)(17,27)(21,31)(23,33)(24,32)(26,34)(28,35)(30,36)(37,38)(39,40);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,26)(13,23)(14,22)(15,24)(17,34)(18,32)(19,33)(20,29)(21,27)(25,40)(28,38)(30,39)(31,35)(36,37);(9,12,13,27,15,28,14,30)(10,17,18,25,19,36,29,35)(11,21,23,38,24,39,22,26)(16,31,32,34,33,40,20,37);(2,6)(5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313427);