#410411

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

803 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317607);