#7003473

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	#42992	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1897 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)(15,24)(16,30)(17,21)(18,34)(19,22)(20,29)(23,25)(26,36)(27,38)(28,42)(31,41)(32,39)(33,40)(35,44)(37,43);(5,6)(7,11)(8,10)(9,12)(13,15,25,43,19,28,18,29)(14,20,34,42,22,37,23,24)(16,31,33,27,17,26,32,44)(21,38,40,41,30,35,39,36);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,26)(17,19)(18,32)(20,35)(22,30)(23,39)(24,41)(25,33)(27,29)(28,31)(34,40)(36,42)(37,38)(43,44);(7,12)(9,11)(13,17,19,16)(14,21,22,30)(15,27,28,44)(18,32,25,33)(20,36,37,41)(23,40,34,39)(24,38,42,35)(26,43,31,29);(6,10)(9,11)(14,22)(15,28)(16,17)(23,34)(29,43)(32,33)(35,38)(36,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,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,d^c=dh,e^c=ei,e^d=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, 6910669);