#7459741

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1328 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)(3,6)(4,5)(7,8)(25,26,32,36,31,37,30,47)(27,38,43,54,42,33,41,50)(28,44,45,51,29,35,46,34)(39,49,55,48,40,53,56,52);(2,5)(3,7)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(25,27,31,42)(26,33,37,38)(28,39,29,40)(30,41,32,43)(34,48,51,52)(35,49,44,53)(36,50,47,54)(45,55,46,56);(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,28,31,29)(26,34,37,51)(27,39,42,40)(30,45,32,46)(33,48,38,52)(35,47,44,36)(41,55,43,56)(49,54,53,50);(1,3,4,7)(2,6,5,8)(10,15)(11,18)(16,20)(19,24)(25,29,31,28)(26,35,37,44)(27,40,42,39)(30,45,32,46)(33,49,38,53)(34,47,51,36)(41,55,43,56)(48,54,52,50);(2,5)(6,8)(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(25,30,31,32)(26,36,37,47)(27,41,42,43)(28,45,29,46)(33,50,38,54)(34,35,51,44)(39,55,40,56)(48,49,52,53)

Polycyclic presentation:

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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7366937);