#394959

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	#23402	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1540 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,26)(13,19)(14,21)(15,20)(17,27)(18,28)(22,31)(23,39)(24,36)(25,33)(29,32)(30,35)(34,37)(38,40);(2,5)(6,8)(9,11)(10,16)(12,23)(13,25)(14,29)(15,24)(17,34)(18,31)(19,33)(20,36)(21,32)(22,28)(26,39)(27,37)(30,38)(35,40);(9,12)(10,17)(11,22)(13,27)(14,30)(15,28)(16,31)(18,21)(19,26)(20,35)(23,29)(24,37)(25,38)(32,39)(33,40)(34,36);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,23)(13,27)(14,28)(15,30)(16,31)(17,21)(19,35)(20,26)(22,29)(24,38)(25,37)(32,34)(33,40)(36,39)

Polycyclic presentation:

2^9|d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,g^c=gi,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, 302155);