#6891206

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1344 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)(11,19)(12,16)(13,18)(14,17)(15,22)(20,23)(21,24)(25,26,29,34)(27,36,38,31)(28,40,42,33)(30,45,44,35)(32,46,51,43)(37,48,41,52)(39,49,55,53)(47,54,50,56);(1,2)(3,7)(4,6)(5,8)(9,11,14,22)(10,15,17,19)(12,20,18,24)(13,23,16,21)(25,27)(26,31)(28,41)(29,38)(30,39)(32,50)(33,52)(34,36)(35,49)(37,42)(40,48)(43,56)(44,55)(45,53)(46,54)(47,51);(1,3)(2,5)(4,8)(6,7)(9,12,14,18)(10,16,17,13)(11,20,22,24)(15,21,19,23)(25,28)(26,32)(27,37)(29,42)(30,46)(31,47)(33,35)(34,51)(36,50)(38,41)(39,56)(40,45)(43,44)(48,49)(52,53)(54,55);(3,8)(5,7)(10,17)(12,18)(15,19)(20,24)(25,28)(26,33)(27,37)(29,42)(30,43)(31,48)(32,35)(34,40)(36,52)(38,41)(39,54)(44,46)(45,51)(47,49)(50,53)(55,56);(2,6)(5,7)(9,13,14,16)(10,18,17,12)(11,21,22,23)(15,20,19,24)(27,38)(28,42)(31,36)(32,51)(33,40)(39,55)(43,46)(49,53)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6798402);