#419822

order	512
nilpotency class	3
isoclinism	isoclinic class of #419454
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1589 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)(7,17,20,12)(8,13)(9,23,18,28)(10,16)(11,15)(14,24,33,36)(19,31)(21,35,25,30)(22,32)(26,29)(27,34);(1,2)(3,4)(5,7)(6,12)(8,18)(9,19)(10,25)(11,20)(13,23)(14,32)(15,17)(16,35)(21,27)(22,24)(26,36)(28,31)(29,33)(30,34);(2,4)(5,8,11,19)(6,13,15,31)(7,18,20,9)(10,26,27,22)(12,28,17,23)(14,30,33,35)(16,29,34,32)(21,24,25,36);(5,9,11,18)(6,14)(7,19,20,8)(10,24,27,36)(12,29)(13,30)(15,33)(16,23)(17,32)(21,26,25,22)(28,34)(31,35)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=ei,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,h^b=hi,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, 327018);