#472007

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1390 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)(7,15,17,11)(8,20,21,35)(9,13)(10,14)(12,22,26,30)(16,23)(18,32)(19,28,29,25)(24,33)(27,36)(31,34);(1,2,3,4)(5,7,16,29,9,17,10,19)(6,11,23,28,13,15,14,25)(8,18,30,36,21,31,22,33)(12,24,35,32,26,27,20,34);(2,4)(7,17)(12,26)(18,31)(19,29)(20,35)(24,27)(32,34)(33,36);(5,8)(6,12,13,26)(7,18)(9,21)(10,22)(11,24,15,27)(14,20,23,35)(16,30)(17,31)(19,33)(25,34,28,32)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379203);