#359375

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

766 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265431, #72307328, #72307361
maximal normal subgroup of	#72265431, #72307328, #72307361

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266571);