#6994361

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2146 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,17)(15,23,27,25)(16,21,28,31)(18,32,34,22)(20,33,30,26)(24,37,38,40)(29,42,41,36)(35,43,44,39);(5,6)(7,11)(8,10)(9,12)(13,15,19,27)(14,20,17,30)(16,24,28,38)(18,33,34,26)(21,35,31,44)(22,25,32,23)(29,43,41,39)(36,40,42,37);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,28)(14,21,17,31)(15,24,27,38)(18,29,34,41)(20,35,30,44)(22,36,32,42)(23,37,25,40)(26,39,33,43);(7,12)(9,11)(13,17,19,14)(15,25,27,23)(16,21,28,31)(18,22,34,32)(20,26,30,33)(24,37,38,40)(29,42,41,36)(35,43,44,39);(6,10)(9,11)(16,28)(21,31)(24,38)(29,41)(35,44)(36,42)(37,40)(39,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6901557);