#6498075

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

1221 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,36)(28,41)(29,33)(30,44)(31,35)(32,40)(34,45)(37,51)(38,49)(39,50)(42,48)(43,56)(46,53)(47,54)(52,55);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,30,40,31,39,45,54)(26,32,34,36,35,47,44,50)(28,37,42,52,29,38,43,53)(33,46,48,49,41,55,56,51);(2,5)(6,8)(11,17)(14,21)(18,23)(22,24)(25,28,31,29)(26,33,35,41)(27,37,39,38)(30,42,45,43)(32,46,47,55)(34,48,44,56)(36,49,50,51)(40,52,54,53);(1,3,4,7)(2,6,5,8)(11,17)(14,21)(18,23)(22,24)(25,29,31,28)(26,33,35,41)(27,38,39,37)(30,43,45,42)(32,46,47,55)(34,48,44,56)(36,49,50,51)(40,53,54,52);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(27,39)(28,29)(32,47)(33,41)(36,50)(40,54)(42,43)(48,56)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,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, 6405271);