#6442570

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

4580 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)(7,16)(8,13)(9,14)(10,22)(11,15)(12,18)(17,32)(19,27)(20,29)(21,31)(23,25)(24,30)(26,33)(28,34)(35,36);(5,7,11,19)(6,12,15,25)(8,17,21,35)(9,18,10,23)(13,28,31,26)(14,27,22,16)(20,33,24,34)(29,32,30,36);(1,2,3,4)(9,10)(12,25)(14,22)(18,23)(20,24)(26,28)(29,30)(33,34);(5,8)(6,13)(7,17)(9,20)(10,24)(11,21)(12,26)(14,29)(15,31)(16,32)(18,33)(19,35)(22,30)(23,34)(25,28)(27,36);(5,9,11,10)(6,14,15,22)(7,18,19,23)(8,20,21,24)(12,27,25,16)(13,29,31,30)(17,33,35,34)(26,36,28,32)

Polycyclic presentation:

2^9|b^2=i,c^2=g,e^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	#72341765, #72341788
maximal normal subgroup of	#72341765, #72341788

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349766);