#7497551

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1313 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)(3,6)(4,5)(7,8)(10,14)(12,20)(15,17)(19,24)(25,26,30,36)(27,38,41,32)(28,42,45,34)(29,37,31,35)(33,47,52,44)(39,49,43,53)(40,50,46,54)(48,56,51,55);(2,5)(3,7)(9,10,13,14)(11,17,18,15)(12,21,20,16)(19,22,24,23)(25,27,30,41)(26,32,36,38)(28,43,45,39)(29,46,31,40)(33,51,52,48)(34,53,42,49)(35,54,37,50)(44,55,47,56);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,30,45)(26,33,36,52)(27,39,41,43)(29,44,31,47)(32,48,38,51)(34,35,42,37)(40,55,46,56)(49,54,53,50);(1,3,4,7)(2,6,5,8)(11,18)(15,17)(19,24)(22,23)(25,28)(26,34)(27,39)(29,47)(30,45)(31,44)(32,49)(33,37)(35,52)(36,42)(38,53)(40,56)(41,43)(46,55)(48,50)(51,54);(2,5)(6,8)(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(25,29,30,31)(26,35,36,37)(27,40,41,46)(28,44,45,47)(32,50,38,54)(33,42,52,34)(39,55,43,56)(48,49,51,53)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,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, 7404747);