#320695

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

671 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=gh,c^2=hi,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 227891);