#419922

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

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

1564 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=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, 327118);