#418824

order	512
nilpotency class	3
isoclinism	isoclinic class of #417704
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	#60368	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1917 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)(7,17)(8,21)(9,19)(10,15,28,33)(12,14)(13,24)(18,23,22,29)(20,27)(25,32)(26,31)(30,34,35,36);(1,2)(3,4)(5,7)(6,12)(8,22)(9,16)(10,14)(11,20)(13,23)(15,27)(17,33)(18,26)(19,28)(21,36)(24,35)(25,30)(29,32)(31,34);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,26)(12,29)(14,30)(15,31)(16,32)(17,34)(19,35)(20,22)(21,33)(25,28)(27,36);(5,9,11,12)(6,14,16,19)(7,15,20,33)(8,23,26,29)(10,17,28,27)(13,30,32,35)(18,31,22,21)(24,34,25,36)

Polycyclic presentation:

2^9|a^2=ef,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,e^d=ei

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, 326020);