#6437280

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433044
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

1244 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,e^b=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, 6344476);