#6994293

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	#45245	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2168 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)(15,23,26,25)(16,21,29,28)(18,32,34,22)(20,33,31,27)(24,37,40,38)(30,42,43,36)(35,44,41,39);(5,6)(7,11)(8,10)(9,12)(13,15,19,26)(14,20,17,31)(16,24,29,40)(18,33,34,27)(21,35,28,41)(22,25,32,23)(30,44,43,39)(36,38,42,37);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,24)(17,28)(18,30)(19,29)(20,35)(22,36)(23,37)(25,38)(26,40)(27,39)(31,41)(32,42)(33,44)(34,43);(7,12)(9,11)(13,17,19,14)(15,25,26,23)(16,28,29,21)(18,22,34,32)(20,27,31,33)(24,38,40,37)(30,36,43,42)(35,39,41,44);(6,10)(9,11)(15,26)(16,29)(20,31)(21,28)(23,25)(27,33)(30,43)(36,42)

Polycyclic presentation:

2^9|a^2=i,b^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=ehi,f^b=fi,d^c=dh,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, 6901489);