#6443329

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437963
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4953 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311821, #72312230, #72313645, …
maximal normal subgroup of	#72311821, #72312230, #72313645, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350525);