#419778

order	512
nilpotency class	3
isoclinism	isoclinic class of #417665
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1572 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,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, 326974);