#394554

order	512
nilpotency class	3
isoclinism	isoclinic class of #394020
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35676	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2218 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:

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

Polycyclic presentation:

2^9|c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi,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, 301750);