#7498572

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1297 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=i,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7405768);