#439370

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1204 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efg,c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,e^d=ei,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, 346566);