#6442568

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	#62848	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4484 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)(7,17)(8,13)(9,14)(10,24)(11,16)(12,19)(15,26)(18,34)(20,30)(21,31)(22,33)(23,28)(25,32)(27,36)(29,35);(5,7,11,9)(6,12,16,28)(8,18,21,20)(10,23,26,19)(13,29,31,27)(14,24,17,15)(22,36,25,35)(30,33,34,32);(1,2,3,4)(5,8)(6,13)(7,18)(9,20)(10,25)(11,21)(12,27)(14,30)(15,33)(16,31)(17,34)(19,36)(22,26)(23,35)(24,32)(28,29);(5,9,11,7)(6,14,16,17)(8,20,21,18)(10,19,26,23)(12,15,28,24)(13,30,31,34)(22,35,25,36)(27,33,29,32);(8,21)(13,31)(18,20)(22,25)(27,29)(30,34)(32,33)(35,36)

Polycyclic presentation:

2^9|b^2=i,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	#72341707, #72341765
maximal normal subgroup of	#72341707, #72341765

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349764);