#7148421

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

1747 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)(16,26)(17,29)(20,32)(21,27)(22,30)(23,37)(25,40)(33,36)(34,38)(35,41)(43,44);(6,9)(7,11)(13,14,18,22)(15,20,24,35)(16,27,26,33)(17,30,29,19)(21,31,36,28)(23,38,37,43)(25,41,40,32)(34,42,44,39);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,24)(14,20,22,35)(16,23,26,37)(17,25,29,40)(19,32,30,41)(21,34,36,44)(27,38,33,43)(28,39,31,42);(6,9)(10,12)(13,16,18,26)(14,21,22,36)(15,23,24,37)(17,31,29,28)(19,33,30,27)(20,34,35,44)(25,42,40,39)(32,43,41,38);(5,7,8,11)(6,10,9,12)(15,24)(20,35)(23,37)(25,40)(32,41)(34,44)(38,43)(39,42)

Polycyclic presentation:

2^9|a^2=g,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, 7055617);