#6672774

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi

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, 6579970);