#7149355

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	C₂×#3221	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ 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)(7,11)(10,12)(13,14)(15,23)(16,20)(17,31)(18,21)(19,27)(22,24)(25,37)(26,34)(28,33)(29,36)(30,42)(32,41)(35,38)(39,44)(40,43);(6,9)(7,11)(13,15,24,26,18,28,17,27)(14,19,31,33,21,34,22,23)(16,25,38,39,29,40,30,41)(20,32,42,43,36,44,35,37);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,29)(14,20,21,36)(15,25,28,40)(17,30,24,38)(19,32,34,44)(22,35,31,42)(23,37,33,43)(26,39,27,41);(6,9)(10,12)(15,26)(17,24)(19,33)(22,31)(23,34)(25,39)(27,28)(30,38)(32,43)(35,42)(37,44)(40,41);(5,7,8,11)(6,10,9,12)(14,21)(16,29)(19,34)(22,31)(23,33)(25,40)(30,38)(39,41)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7056551);