#6766445

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682596
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1370 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)(3,5)(4,6)(7,8)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,31)(28,40)(29,33)(30,35)(32,45)(34,42)(36,53)(37,47)(38,48)(39,51)(41,56)(43,52)(44,50)(46,54)(49,55);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,31)(28,41)(29,39)(30,37)(32,49)(33,51)(34,48)(35,47)(36,43)(38,42)(40,56)(44,54)(45,55)(46,50)(52,53);(2,6)(3,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,42,44,30,43,29,45)(26,32,33,52,35,50,34,40)(27,36,38,55,37,41,39,54)(31,46,51,56,47,49,48,53);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(27,37)(28,43)(31,47)(32,50)(38,39)(40,52)(44,45)(48,51);(3,7)(5,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29,30,42)(26,33,35,34)(27,38,37,39)(28,44,43,45)(31,48,47,51)(32,40,50,52)(36,54,41,55)(46,56,49,53)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fgh,d^2=f,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,d^c=di,e^c=ei

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, 6673641);