#7054709

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2084 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)(16,21,29,28)(18,32,34,22)(20,33,31,26)(24,37,39,38)(30,42,43,36)(35,44,41,40);(5,6)(7,11)(8,10)(9,12)(13,15,19,27)(14,20,17,31)(16,24,29,39)(18,33,34,26)(21,35,28,41)(22,25,32,23)(30,44,43,40)(36,38,42,37);(2,4)(5,7,8,12)(6,9,10,11)(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)(15,25,27,23)(16,28,29,21)(18,22,34,32)(20,26,31,33)(24,38,39,37)(30,36,43,42)(35,40,41,44);(6,10)(9,11)(16,29)(21,28)(24,39)(30,43)(35,41)(36,42)(37,38)(40,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,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=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, 6961905);