#6774455

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1189 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6681651);