#6437936

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437935
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	#62853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4615 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:

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

Polycyclic presentation:

2^9|a^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345132);