#7497583

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

1311 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)(27,37,40,32)(28,34,43,52)(29,35,31,46)(33,45,53,42)(38,54,41,48)(39,50,44,49)(47,56,51,55);(2,5)(3,7)(9,10,13,14)(11,17,18,15)(12,21,20,16)(19,22,24,23)(25,27,30,40)(26,32,36,37)(28,41,43,38)(29,44,31,39)(33,51,53,47)(34,54,52,48)(35,50,46,49)(42,55,45,56);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,30,43)(26,33,36,53)(27,38,40,41)(29,42,31,45)(32,47,37,51)(34,35,52,46)(39,55,44,56)(48,50,54,49);(1,3,4,7)(2,6,5,8)(11,18)(15,17)(19,24)(22,23)(25,28)(26,34)(27,38)(29,45)(30,43)(31,42)(32,48)(33,46)(35,53)(36,52)(37,54)(39,56)(40,41)(44,55)(47,49)(50,51);(2,5)(6,8)(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(25,29,30,31)(26,35,36,46)(27,39,40,44)(28,42,43,45)(32,49,37,50)(33,52,53,34)(38,55,41,56)(47,48,51,54)

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,e^a=efi,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, 7404779);