#6437980

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
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 ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4540 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,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, 6345176);