#7470867

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1383 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)(11,17)(14,18)(19,23)(22,24)(25,26,29,35,30,36,31,37)(27,38,45,49,41,32,40,54)(28,34,46,33,44,53,43,51)(39,52,55,50,42,48,56,47);(2,5)(3,7)(9,10)(11,18)(12,20)(13,15)(14,17)(16,21)(19,22)(23,24)(25,27,30,41)(26,32,36,38)(28,42,44,39)(29,45,31,40)(33,50,51,47)(34,52,53,48)(35,54,37,49)(43,55,46,56);(9,11)(10,14)(12,19)(13,17)(15,18)(16,22)(20,24)(21,23)(25,28,30,44)(26,33,36,51)(27,39,41,42)(29,43,31,46)(32,47,38,50)(34,37,53,35)(40,55,45,56)(48,49,52,54);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(16,20)(19,23)(25,28)(26,34)(27,39)(29,46)(30,44)(31,43)(32,48)(33,35)(36,53)(37,51)(38,52)(40,56)(41,42)(45,55)(47,54)(49,50);(2,5)(6,8)(9,12,13,21)(10,16,15,20)(11,19,17,23)(14,22,18,24)(25,29,30,31)(26,35,36,37)(27,40,41,45)(28,43,44,46)(32,49,38,54)(33,34,51,53)(39,55,42,56)(47,52,50,48)

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,c^2=i,d^2=fi,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, 7378063);