#7182151

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^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,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, 7089347);