#416260

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

898 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)(7,19,23,31)(8,24,28,33)(9,29)(10,16)(11,17)(12,18)(13,21,32,22)(14,26,34,27)(15,20)(25,35)(30,36);(5,7,12,23)(6,13)(8,25,28,9)(10,21,11,22)(14,29)(15,24)(16,19)(17,31)(18,32)(20,26,30,27)(33,36)(34,35);(2,4)(5,8)(6,14)(7,20)(9,22)(10,26)(11,27)(12,28)(13,15)(16,24)(17,33)(18,34)(19,35)(21,25)(23,30)(29,31)(32,36);(5,9,10,20,12,25,11,30)(6,15,17,29,18,36,16,35)(7,8,21,26,23,28,22,27)(13,24,31,14,32,33,19,34)

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=di,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, 323456);