#7497555

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1336 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)(10,14)(12,20)(15,17)(19,24)(25,26,29,34)(27,36,38,31)(28,40,41,33)(30,32,43,35)(37,45,50,49)(39,46,52,44)(42,48,53,47)(51,54,56,55);(2,5)(3,7)(9,10,13,14)(11,17,18,15)(12,21,20,16)(19,22,24,23)(25,27,29,38)(26,31,34,36)(28,37,41,50)(30,39,43,52)(32,44,35,46)(33,45,40,49)(42,51,53,56)(47,54,48,55);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(26,32)(30,43)(31,44)(33,47)(34,35)(36,46)(39,52)(40,48)(42,53)(45,54)(49,55)(51,56);(1,3,4,7)(2,6,5,8)(11,18)(15,17)(19,24)(22,23)(25,28,29,41)(26,33,34,40)(27,37,38,50)(30,42,43,53)(31,45,36,49)(32,47,35,48)(39,51,52,56)(44,54,46,55);(2,5)(6,8)(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(26,34)(28,41)(31,36)(32,35)(37,50)(42,53)(44,46)(51,56)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,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, 7404751);