#96663

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#376
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

302 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 56 points:

(2,5)(6,8)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24)(25,26)(27,35,38,48)(28,32)(29,33)(30,34)(31,37,43,51)(36,47,50,55)(39,44)(40,45)(41,46)(42,49,53,56)(52,54);(1,2,4,6,3,5,7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,31,34,43)(28,36)(29,37)(30,38)(32,42,45,53)(33,35,46,48)(39,49)(40,50)(41,51)(44,47,54,55)(52,56);(11,17)(14,20)(18,23)(21,24)(25,28,29,39,30,40,41,52)(26,32,33,44,34,45,46,54)(27,36,37,49,38,50,51,56)(31,42,35,47,43,53,48,55)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=di,d^2=g,e^2=h,f^2=i,b^a=bd,f^a=fg,c^b=ch,e^b=eg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3859);