#7199897

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1273 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=gi,d^2=fh,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7107093);