#6916645

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1243 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)(27,31)(28,38)(29,42)(30,33)(32,41)(34,44)(35,51)(36,46)(37,54)(39,56)(40,50)(43,49)(45,52)(47,53)(48,55);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,30,36)(26,31,33,46)(28,39,40,35)(29,37,44,53)(32,48,49,45)(34,47,42,54)(38,56,50,51)(41,55,43,52);(2,5)(6,8)(12,21)(15,20)(18,24)(22,23)(25,28,30,40)(26,32,33,49)(27,35,36,39)(29,43,44,41)(31,45,46,48)(34,50,42,38)(37,55,53,52)(47,56,54,51);(1,3,4,7)(2,6,5,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(27,36)(28,40)(31,46)(32,49)(37,53)(38,50)(41,43)(47,54);(11,19)(14,17)(18,24)(22,23)(26,33)(27,36)(32,49)(34,42)(35,39)(37,53)(38,50)(52,55)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6823841);