#403013

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	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1219 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)(5,6,10,23,11,8,21,14)(7,15,19,16,20,12,18,28)(9,13,25,30,24,22,26,29)(17,27,35,32,33,31,34,36);(5,7,10,18,11,20,21,19)(6,12,23,16,8,15,14,28)(9,17,26,35,24,33,25,34)(13,27,29,32,22,31,30,36);(5,8)(6,11)(7,16)(9,22)(10,14)(12,19)(13,24)(15,18)(17,32)(20,28)(21,23)(25,29)(26,30)(27,34)(31,35)(33,36);(2,4)(5,9,10,26,11,24,21,25)(6,13,23,29,8,22,14,30)(7,17,18,35,20,33,19,34)(12,27,16,32,15,31,28,36)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,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	#72250020, #72257160, #72257201
maximal normal subgroup of	#72250020, #72257160, #72257201

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310209);