#6808178

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#42788	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1835 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=fi,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6715374);