#6889855

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1341 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6797051);