#394948

order	512
nilpotency class	3
isoclinism	isoclinic class of #394113
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×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1473 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:

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

Polycyclic presentation:

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

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, 302144);