#6776991

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1250 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi

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, 6684187);