#6437943

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	#62859	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4556 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,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.

Relationships

maximal subgroup of	#72312395, #72312619, #72313601, …
maximal normal subgroup of	#72312395, #72312619, #72313601, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345139);