#6443336

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

4943 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)(5,6,11,15)(7,16,14,10)(8,12,20,27)(9,13,23,28)(17,21,29,24)(18,22,30,25)(19,26,32,35)(31,34,36,33);(5,7,10,6,11,14,16,15)(8,17,21,27,20,29,24,12)(9,18,22,28,23,30,25,13)(19,31,33,26,32,36,34,35);(1,2,3,4)(5,8)(6,12)(7,17)(9,19)(10,24)(11,20)(13,26)(14,29)(15,27)(16,21)(18,31)(22,34)(23,32)(25,33)(28,35)(30,36);(5,9,11,23)(6,13,15,28)(7,18,14,30)(8,19,20,32)(10,25,16,22)(12,26,27,35)(17,31,29,36)(21,34,24,33);(8,20)(12,27)(17,29)(19,32)(21,24)(26,35)(31,36)(33,34)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=i,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	#72341700, #72341707, #72341753
maximal normal subgroup of	#72341700, #72341707, #72341753

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350532);