#6607197

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,19,24)(15,25,27,37)(16,30,34,39)(17,23,18,22)(20,29,36,28)(21,32,40,33)(26,35,31,38)(41,42,44,43);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,28,19,27,18,29)(14,20,22,25,24,36,23,37)(16,31,32,43,34,26,33,42)(21,38,39,44,40,35,30,41);(7,12)(9,10)(13,16,17,32,19,34,18,33)(14,21,22,39,24,40,23,30)(15,26,28,42,27,31,29,43)(20,35,25,41,36,38,37,44);(5,7)(6,10)(8,12)(9,11)(15,27)(20,36)(25,37)(26,31)(28,29)(35,38)(41,44)(42,43);(7,12)(9,10)(15,27)(20,36)(25,37)(26,31)(28,29)(35,38)(41,44)(42,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=gi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^d=ehi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6514393);