#357889

order	512
nilpotency class	3
isoclinism	isoclinic class of #9109
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2642 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)(7,15,18,11)(8,19)(9,13)(10,14)(12,21)(16,25,30,23)(17,28,31,24)(20,32)(22,27)(26,33)(29,35,36,34);(1,2)(3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,25)(13,24)(14,15)(19,23)(20,29)(21,30)(22,31)(26,35)(27,28)(32,34)(33,36);(2,4)(5,8)(6,12,14,19)(7,16)(9,20)(10,21)(11,23,15,25)(13,26,27,32)(17,29)(18,30)(22,33)(24,34,28,35)(31,36);(5,9,10,22)(6,13,14,27)(7,17,18,31)(8,20,21,33)(11,24,15,28)(12,26,19,32)(16,29,30,36)(23,34,25,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314370, #72314393, #72314414
maximal normal subgroup of	#72314370, #72314393, #72314414

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265085);