#303759

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

780 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=ef,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 210955);