#7148437

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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1765 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)(14,19)(15,23)(16,26)(17,29)(20,34)(21,27)(22,30)(32,36)(33,37)(35,43)(40,44)(41,42);(6,9)(7,11)(13,14,18,22)(15,20,23,36)(16,27,26,33)(17,30,29,19)(21,31,37,28)(24,40,38,43)(25,34,39,32)(35,42,44,41);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,23)(14,20,22,36)(16,24,26,38)(17,25,29,39)(19,32,30,34)(21,35,37,44)(27,40,33,43)(28,41,31,42);(6,9)(10,12)(13,16,18,26)(14,21,22,37)(15,24,23,38)(17,31,29,28)(19,33,30,27)(20,35,36,44)(25,42,39,41)(32,43,34,40);(5,7,8,11)(6,10,9,12)(15,23)(20,36)(24,38)(25,39)(32,34)(35,44)(40,43)(41,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7055633);