#6369940

order	512
nilpotency class	3
isoclinism	isoclinic class of #3585
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₄⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226779, #72226797
maximal normal subgroup of	#72226779, #72226797

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277136);