#7199914

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=g,d^2=fhi,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7107110);