#400478

order	512
nilpotency class	3
isoclinism	isoclinic class of #394469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1019 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321973
maximal normal subgroup of	#72321973

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307674);