#7497554

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

1327 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)(25,26,32,37)(27,33,41,50)(28,35,29,43)(30,36,31,46)(34,45,51,42)(38,52,44,48)(39,49,40,55)(47,54,56,53);(2,5)(3,7)(9,10,13,14)(11,17,18,15)(12,21,20,16)(19,22,24,23)(25,27)(26,33)(28,38)(29,44)(30,39)(31,40)(32,41)(34,47)(35,52)(36,49)(37,50)(42,53)(43,48)(45,54)(46,55)(51,56);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,32,29)(26,34,37,51)(27,38,41,44)(30,42,31,45)(33,47,50,56)(35,36,43,46)(39,53,40,54)(48,55,52,49);(1,3,4,7)(2,6,5,8)(11,18)(15,17)(19,24)(22,23)(25,29,32,28)(26,35,37,43)(27,38,41,44)(30,42,31,45)(33,48,50,52)(34,36,51,46)(39,54,40,53)(47,55,56,49);(2,5)(6,8)(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(25,30,32,31)(26,36,37,46)(27,39,41,40)(28,42,29,45)(33,49,50,55)(34,43,51,35)(38,53,44,54)(47,48,56,52)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=fhi,e^2=hi,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, 7404750);