#6483497

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470901
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	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2028 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)(18,35)(19,24)(20,28)(21,27)(23,36)(26,41)(29,39)(31,37)(32,40)(33,38)(34,44)(42,43);(1,2,3,4)(7,8)(10,12)(13,15,18,28,19,29,36,33)(14,20,23,25,24,38,35,39)(16,26,27,43,31,17,32,34)(21,22,30,42,40,41,37,44);(5,7,9,8)(6,10,11,12)(13,16,18,27,19,31,36,32)(14,21,23,30,24,40,35,37)(15,26,28,43,29,17,33,34)(20,22,25,42,38,41,39,44);(5,8,9,7)(6,10,11,12)(13,17,19,26)(14,22,24,41)(15,27,29,32)(16,28,31,33)(18,34,36,43)(20,37,38,30)(21,39,40,25)(23,42,35,44);(6,11)(10,12)(16,31)(17,26)(21,40)(22,41)(27,32)(30,37)(34,43)(42,44)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=hi,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,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6390693);