#6917192

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885065
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1407 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=i,d^2=fi,e^2=i,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=dfh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6824388);