#7063081

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#42797	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1855 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)(14,19)(15,23)(17,30)(20,34)(21,26)(22,31)(24,27)(28,32)(33,36)(35,44)(37,41)(38,39);(5,6)(7,11)(8,10)(9,12)(13,14,18,22)(15,20,23,36)(16,26,29,39)(17,31,30,19)(21,28,38,32)(24,41,27,44)(25,34,40,33)(35,42,37,43);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,27)(17,25)(18,23)(19,33)(21,37)(22,36)(24,29)(26,44)(28,43)(30,40)(31,34)(32,42)(35,38)(39,41);(7,12)(9,11)(13,16,18,29)(14,21,22,38)(15,24,23,27)(17,32,30,28)(19,26,31,39)(20,35,36,37)(25,43,40,42)(33,41,34,44);(6,10)(9,11)(14,22)(15,23)(19,31)(21,38)(24,27)(25,40)(26,39)(42,43)

Polycyclic presentation:

2^9|a^2=gi,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=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6970277);