#7470849

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

1386 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)(25,26,30,37,31,36,46,35)(27,38,41,50,42,32,47,51)(28,34,44,52,29,45,43,33)(39,54,56,48,40,49,55,53);(2,5)(3,7)(9,10)(11,18)(12,20)(13,15)(14,17)(16,21)(19,22)(23,24)(25,27,31,42)(26,32,36,38)(28,39,29,40)(30,41,46,47)(33,48,52,53)(34,49,45,54)(35,50,37,51)(43,55,44,56);(9,11)(10,14)(12,19)(13,17)(15,18)(16,22)(20,24)(21,23)(25,28,31,29)(26,33,36,52)(27,39,42,40)(30,43,46,44)(32,48,38,53)(34,37,45,35)(41,55,47,56)(49,51,54,50);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(16,20)(19,23)(25,29,31,28)(26,34,36,45)(27,40,42,39)(30,43,46,44)(32,49,38,54)(33,37,52,35)(41,55,47,56)(48,51,53,50);(2,5)(6,8)(9,12,13,21)(10,16,15,20)(11,19,17,23)(14,22,18,24)(25,30,31,46)(26,35,36,37)(27,41,42,47)(28,43,29,44)(32,50,38,51)(33,34,52,45)(39,55,40,56)(48,49,53,54)

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,c^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,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, 7378045);