#6660637

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660434
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1442 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,32,38)(27,33,41,50)(28,42,29,35)(30,37,31,36)(34,45,52,44)(39,48,43,56)(40,53,46,49)(47,54,51,55);(3,7)(5,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,33)(28,43)(29,39)(30,46)(31,40)(32,41)(34,51)(35,48)(36,53)(37,49)(38,50)(42,56)(44,54)(45,55)(47,52);(1,3)(2,5)(4,7)(6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28)(26,34)(27,39)(29,32)(30,44)(31,45)(33,47)(35,37)(36,42)(38,52)(40,54)(41,43)(46,55)(48,53)(49,56)(50,51);(3,7)(5,8)(10,16)(12,20)(14,17)(18,24)(25,29,32,28)(26,35,38,42)(27,39,41,43)(30,44,31,45)(33,48,50,56)(34,36,52,37)(40,55,46,54)(47,49,51,53);(2,6)(5,8)(10,16)(14,17)(15,22)(21,23)(25,30,32,31)(26,36,38,37)(27,40,41,46)(28,44,29,45)(33,49,50,53)(34,42,52,35)(39,54,43,55)(47,56,51,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,e^b=ei,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6567833);