#6498024

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1312 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)(3,7)(4,5)(6,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,30,34)(27,36,40,53)(28,33,42,32)(29,44,37,35)(31,45,48,39)(38,52,41,51)(43,49,54,50)(46,55,47,56);(2,5)(6,7)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,37,45,30,40,29,39)(26,31,44,53,34,48,35,36)(28,41,54,55,42,38,43,56)(32,47,50,51,33,46,49,52);(1,3,4,8)(2,6,5,7)(11,17)(14,21)(18,23)(22,24)(25,28)(26,32)(27,38)(29,43)(30,42)(31,46)(33,34)(35,49)(36,51)(37,54)(39,55)(40,41)(44,50)(45,56)(47,48)(52,53);(11,17)(14,21)(18,23)(22,24)(25,28)(26,33)(27,38)(29,43)(30,42)(31,47)(32,34)(35,50)(36,52)(37,54)(39,55)(40,41)(44,49)(45,56)(46,48)(51,53);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,34)(28,42)(31,48)(35,44)(36,53)(38,41)(43,54)(55,56)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6405220);