#6369765

order	512
nilpotency class	3
isoclinism	isoclinic class of #2883
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂²	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

7478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=i,g^2=i,b^a=bf,f^a=fi,g^b=gi,d^c=di

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316937, #72316938, #72316945
maximal normal subgroup of	#72316937, #72316938, #72316945

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276961);