#405480

order	512
nilpotency class	3
isoclinism	isoclinic class of #394275
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1425 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)(5,6,11,9)(7,12,18,16)(8,19,23,30)(10,24,26,14)(13,25,21,22)(15,31,20,36)(17,33,35,28)(27,34,29,32);(1,2,4,3)(5,7)(6,12)(8,20,23,15)(9,16)(10,17)(11,18)(13,29,21,27)(14,28)(19,36,30,31)(22,34,25,32)(24,33)(26,35);(1,3,4,2)(5,8)(6,13)(7,15)(9,21)(10,25)(11,23)(12,27)(14,30)(16,29)(17,34)(18,20)(19,24)(22,26)(28,36)(31,33)(32,35);(5,9,11,6)(7,16,18,12)(8,21,23,13)(10,14,26,24)(15,29,20,27)(17,28,35,33)(19,25,30,22)(31,34,36,32)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,f^c=fi,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, 312676);