#6442567

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437452
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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

4546 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)(7,12)(8,13)(9,11)(10,15)(14,21)(16,27)(17,28)(18,26)(19,29)(20,30)(22,24)(23,25)(31,36)(32,34)(33,35);(6,11)(9,21)(12,26)(13,29)(14,15)(18,22)(19,23)(24,28)(25,30)(27,34)(32,33)(35,36);(1,2,3,4)(5,7)(6,12)(8,16)(9,22)(10,17)(11,24)(13,27)(14,26)(15,28)(18,21)(19,33)(20,31)(23,32)(25,34)(29,35)(30,36);(5,8,10,20)(6,13,15,30)(7,16,17,31)(9,23,21,19)(11,25,14,29)(12,27,28,36)(18,33,22,32)(24,34,26,35);(7,17)(12,28)(16,31)(18,22)(24,26)(27,36)(32,33)(34,35)

Polycyclic presentation:

2^9|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,f^b=fi,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, #72341765
maximal normal subgroup of	#72341700, #72341765

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349763);