#7149349

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1872 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)(13,14,20,17)(15,24,27,37)(16,28,31,22)(18,23,19,33)(21,34,32,26)(25,42,29,41)(30,40,35,39)(36,44,38,43);(6,9)(7,11)(13,15,19,26,20,27,18,34)(14,21,23,24,17,32,33,37)(16,29,30,43,31,25,35,44)(22,38,40,42,28,36,39,41);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,20,31)(14,22,17,28)(15,25,27,29)(18,35,19,30)(21,36,32,38)(23,40,33,39)(24,41,37,42)(26,44,34,43);(6,9)(10,12)(13,17,20,14)(15,24,27,37)(16,22,31,28)(18,33,19,23)(21,34,32,26)(25,42,29,41)(30,39,35,40)(36,44,38,43);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,23,17,33)(15,26,27,34)(16,30,31,35)(21,37,32,24)(22,39,28,40)(25,43,29,44)(36,41,38,42)

Polycyclic presentation:

2^9|a^2=g,b^2=f,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,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, 7056545);