#6483419

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
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	#33179	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2098 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)(7,12)(8,10)(9,11)(13,14)(15,25)(16,30)(17,22)(18,23)(19,24)(20,29)(21,32)(26,34)(27,36)(28,35)(31,38)(33,37)(39,41)(40,44)(42,43);(1,2,3,4)(7,8)(10,12)(13,15,17,29,19,27,18,28)(14,20,23,25,24,35,22,36)(16,26,31,42,33,40,32,41)(21,34,37,39,38,44,30,43);(5,7,9,8)(6,10,11,12)(13,16,18,32,19,33,17,31)(14,21,22,30,24,38,23,37)(15,26,28,41,27,40,29,42)(20,34,36,43,35,44,25,39);(5,8,9,7)(6,10,11,12)(15,27)(20,35)(25,36)(26,40)(28,29)(34,44)(39,43)(41,42);(6,11)(10,12)(13,17,19,18)(14,22,24,23)(15,28,27,29)(16,31,33,32)(20,25,35,36)(21,30,38,37)(26,41,40,42)(34,39,44,43)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=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, 6390615);