#472004

order	512
nilpotency class	3
isoclinism	isoclinic class of #465214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

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

1410 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)(7,15,19,16)(8,20,23,12)(9,13)(10,14)(11,18,28,26)(17,30)(21,22,35,34)(24,25)(27,31)(29,32)(33,36);(1,2,3,4)(5,7,10,19)(6,11)(8,17,23,32)(9,18,25,26)(12,27)(13,15)(14,28)(16,24)(20,33)(21,29)(22,31,34,36)(30,35);(2,4)(5,6)(7,16)(8,21)(9,24)(10,14)(11,26)(12,22)(13,25)(15,19)(17,27)(18,28)(20,34)(23,35)(29,36)(30,31)(32,33);(5,8,9,22,10,23,25,34)(6,12,13,21,14,20,24,35)(7,17,18,31,19,32,26,36)(11,27,15,29,28,33,16,30)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379200);