#6693947

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1217 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6601143);