#420380

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1333 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)(7,16,19,30)(8,13)(9,12,18,26)(10,14)(11,15)(17,32,34,31)(20,27,33,24)(21,28)(22,29)(23,25)(35,36);(1,2)(3,4)(5,7)(6,12)(8,17)(9,10)(11,19)(13,27)(14,30)(15,26)(16,25)(18,23)(20,21)(22,34)(24,29)(28,31)(32,36)(33,35);(2,4)(5,8,10,21,11,22,23,35)(6,13,14,28,15,29,25,36)(7,17,18,33,19,34,9,20)(12,24,16,31,26,27,30,32);(5,9,23,19,11,18,10,7)(6,12,25,30,15,26,14,16)(8,20,35,34,22,33,21,17)(13,24,36,32,29,27,28,31)

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,h^c=hi,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, 327576);