#394753

order	512
nilpotency class	3
isoclinism	isoclinic class of #394199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1638 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)(7,15)(9,21)(12,25)(13,23)(14,24)(16,31)(17,30)(18,19)(27,28)(29,32)(33,36);(1,2)(3,4)(5,6,9,11,10,14,21,24)(7,12,15,22,17,25,30,34)(8,13,18,23,20,28,19,27)(16,26,29,33,32,35,31,36);(2,4)(5,7,9,15,10,17,21,30)(6,12,11,22,14,25,24,34)(8,16,18,29,20,32,19,31)(13,26,23,33,28,35,27,36);(5,8)(6,13)(7,16)(9,18)(10,20)(11,23)(12,26)(14,28)(15,29)(17,32)(19,21)(22,33)(24,27)(25,35)(30,31)(34,36)

Polycyclic presentation:

2^9|b^2=e,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, 301949);