#6459757

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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×#3002	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3145 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:

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

Polycyclic presentation:

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

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, 6366953);