#6971580

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907909
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1341 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6878776);