#6442562

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437445
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4553 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)(9,21)(10,14)(11,18)(13,23)(16,32)(17,30)(19,31)(20,29)(22,25)(24,28)(26,35)(27,34)(33,36);(5,7)(6,11)(8,16)(9,22)(10,17)(12,27)(13,30)(14,25)(15,21)(18,23)(19,35)(20,33)(24,34)(26,29)(28,36)(31,32);(1,2,3,4)(9,23)(11,25)(13,21)(18,22)(19,24)(26,27)(28,31)(34,35);(5,8,10,20)(6,12,14,29)(7,16,17,33)(9,24,23,19)(11,26,25,27)(13,31,21,28)(15,32,30,36)(18,35,22,34);(7,17)(11,25)(15,30)(16,33)(18,22)(26,27)(32,36)(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,e^b=ei,f^b=fi,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306292, #72312230
maximal normal subgroup of	#72306292, #72312230
maximal quotient of	#72306295

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349758);