#395247

order	512
nilpotency class	3
isoclinism	isoclinic class of #395246
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1774 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:

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

Polycyclic presentation:

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

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, 302443);