#6442566

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
prime radical	#62842	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4694 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)(7,15)(8,12)(9,21)(10,14)(11,17)(13,24)(16,32)(18,30)(19,28)(20,31)(22,26)(23,29)(25,35)(27,34)(33,36);(5,7,10,18)(6,11,14,26)(8,16,19,33)(9,22,24,17)(12,27,28,25)(13,30,21,15)(20,35,23,34)(29,36,31,32);(1,2,3,4)(5,8)(6,12)(7,16)(9,23)(10,19)(11,25)(13,31)(14,28)(15,32)(17,35)(18,33)(20,24)(21,29)(22,34)(26,27)(30,36);(9,24)(11,26)(13,21)(17,22)(20,23)(25,27)(29,31)(34,35);(8,19)(12,28)(16,33)(20,23)(25,27)(29,31)(32,36)(34,35)

Polycyclic presentation:

2^9|b^2=i,c^2=g,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	#72318136, #72318196, #72318491, …
maximal normal subgroup of	#72318136, #72318196, #72318491, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349762);