#394754

order	512
nilpotency class	3
isoclinism	isoclinic class of #394199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,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, 301950);