#6803812

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1227 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)(11,18,20,15)(12,21,13,16)(19,23,22,24)(25,26,30,35,31,36,46,52)(27,37,42,49,41,32,40,50)(28,34,44,51,29,45,43,33)(38,54,55,47,39,48,56,53);(1,2,4,5)(3,7,8,6)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,27)(26,32)(28,38)(29,39)(30,42)(31,41)(33,47)(34,48)(35,50)(36,37)(40,46)(43,56)(44,55)(45,54)(49,52)(51,53);(2,5)(6,7)(25,28,31,29)(26,33,36,51)(27,38,41,39)(30,43,46,44)(32,47,37,53)(34,35,45,52)(40,55,42,56)(48,50,54,49);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,29,31,28)(26,34,36,45)(27,39,41,38)(30,43,46,44)(32,48,37,54)(33,35,51,52)(40,55,42,56)(47,50,53,49);(1,3,4,8)(2,6,5,7)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,30,31,46)(26,35,36,52)(27,40,41,42)(28,43,29,44)(32,49,37,50)(33,45,51,34)(38,55,39,56)(47,48,53,54)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fhi,c^2=i,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,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, 6711008);