#448740

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

694 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)(7,17)(8,13)(9,14)(10,16)(11,15)(12,20)(18,33)(19,27)(21,28)(22,26)(23,30)(24,31)(25,32)(29,35)(34,36);(5,7,10,21)(6,12,16,27)(8,22)(9,19,11,20)(13,29)(14,17,15,28)(18,23)(24,35)(25,34)(26,30)(31,36)(32,33);(5,8,9,23,10,25,11,24)(6,13,15,31,16,32,14,30)(7,18,20,22,21,35,19,34)(12,26,17,33,27,36,28,29);(2,4)(6,14,16,15)(7,19,21,20)(13,30,32,31)(18,34,35,22)

Polycyclic presentation:

2^9|b^2=f,c^2=e,d^2=eg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=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, 355936);