#413888

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1372 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)(7,19)(8,24)(9,15)(10,17)(11,16)(12,18)(13,22)(14,26)(20,30)(21,36)(23,32)(25,34)(27,31)(28,35)(29,33);(2,4)(5,7)(6,13)(8,25)(9,21)(10,29)(11,22)(12,23)(14,34)(15,31)(16,32)(17,19)(18,33)(20,27)(24,30)(26,28)(35,36);(5,8)(6,14)(7,20)(9,22)(10,26)(11,27)(12,21)(13,30)(15,32)(16,24)(17,36)(18,31)(19,34)(23,28)(25,29)(33,35);(5,9,10,20,12,25,11,28)(6,15,17,30,18,34,16,35)(7,21,29,27,23,8,22,26)(13,31,19,24,33,14,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257140, #72257146, #72257167
maximal normal subgroup of	#72257140, #72257146, #72257167

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321084);