#6443333

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5105 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318220, #72318236, #72318324, …
maximal normal subgroup of	#72318220, #72318236, #72318324, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350529);