#394995

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	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1495 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)(11,21)(12,27)(13,18)(14,20)(15,19)(16,24)(17,29)(22,38)(23,31)(25,33)(26,34)(28,36)(30,35)(32,39)(37,40);(2,5)(6,8)(9,11)(10,16)(12,23)(13,24)(14,25)(15,26)(17,32)(18,21)(19,33)(20,34)(22,29)(27,37)(28,40)(30,39)(31,36)(35,38);(9,12)(10,17)(11,22)(13,28)(14,30)(15,29)(16,31)(18,35)(19,36)(20,27)(21,37)(23,25)(24,39)(26,40)(32,34)(33,38);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,23)(13,28)(14,29)(15,30)(16,32)(18,35)(19,27)(20,36)(21,38)(22,25)(24,40)(26,39)(31,34)(33,37)

Polycyclic presentation:

2^9|d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,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, 302191);