#405518

order	512
nilpotency class	3
isoclinism	isoclinic class of #394262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1232 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:

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

Polycyclic presentation:

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

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, 312714);