#419854

order	512
nilpotency class	3
isoclinism	isoclinic class of #419852
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1602 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,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, 327050);