#7182172

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1738 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)(6,10)(8,11)(14,19)(15,23)(17,31)(20,36)(21,35)(22,32)(24,34)(25,28)(27,39)(29,33)(37,44)(38,41);(5,6,9,10)(7,8,12,11)(13,14)(15,24)(16,27)(17,32)(18,22)(19,31)(20,23)(21,29)(25,41)(26,34)(28,44)(30,35)(33,39)(36,40)(37,43)(38,42);(2,4)(5,7)(6,11)(8,10)(9,12)(13,15)(14,20)(16,28)(17,26)(18,23)(19,34)(21,38)(22,24)(25,30)(27,41)(29,43)(31,40)(32,36)(33,42)(35,44)(37,39);(6,10)(7,12)(13,16,18,30)(14,21,22,39)(15,25,23,28)(17,33,31,29)(19,35,32,27)(20,37,24,38)(26,43,40,42)(34,41,36,44);(5,8,9,11)(6,7,10,12)(14,22)(15,23)(19,32)(21,39)(25,28)(26,40)(27,35)(42,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7089368);