#414378

order	512
nilpotency class	3
isoclinism	isoclinic class of #396611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

951 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321574);