#7142961

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	#45248	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1998 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:

(2,4)(7,11)(10,12)(13,14)(15,23)(16,20)(17,29)(18,21)(19,25)(22,31)(24,37)(26,36)(27,35)(28,41)(30,33)(32,39)(34,40)(38,44)(42,43);(1,2)(3,4)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,19,21,33)(16,24,27,38)(17,30,31,25)(20,32,35,43)(22,36,29,23)(28,42,40,39)(34,44,41,37);(2,4)(6,9)(10,12)(13,16,18,27)(14,20,21,35)(15,24,26,38)(17,28,31,40)(19,32,33,43)(22,34,29,41)(23,37,36,44)(25,39,30,42);(5,7,8,11)(6,10,9,12)(14,21)(16,27)(19,33)(22,29)(23,36)(24,38)(28,40)(39,42)

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 7050157);