#6437397

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433057
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1284 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)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,26,29,33)(27,35,37,31)(28,39,41,55)(30,44,43,34)(32,47,49,42)(36,51,40,54)(38,46,53,50)(45,52,48,56);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27)(26,31)(28,40)(29,37)(30,38)(32,48)(33,35)(34,46)(36,41)(39,51)(42,56)(43,53)(44,50)(45,49)(47,52)(54,55);(11,17)(14,20)(18,23)(21,24)(25,28,30,42,29,41,43,47)(26,32,34,39,33,49,44,55)(27,36,38,52,37,40,53,56)(31,45,46,54,35,48,50,51);(1,3,4,7)(2,6,5,8)(10,15)(14,20)(16,22)(21,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,45)(33,35)(34,46)(39,54)(40,41)(42,52)(43,53)(44,50)(47,56)(48,49)(51,55);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(27,37)(28,41)(31,35)(32,49)(38,53)(39,55)(42,47)(46,50)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344593);