#7205502

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1375 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)(3,5)(4,6)(7,8)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26,30,35)(27,31,37,45)(28,33,40,49)(29,32,42,34)(36,44,39,48)(38,43,52,46)(41,47,54,50)(51,55,53,56);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,31)(28,39)(29,38)(30,37)(32,43)(33,48)(34,46)(35,45)(36,40)(41,53)(42,52)(44,49)(47,56)(50,55)(51,54);(3,7)(5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(29,42)(31,43)(33,47)(34,35)(38,52)(41,54)(44,55)(45,46)(48,56)(49,50)(51,53);(2,6)(3,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,36)(29,41)(30,40)(31,44)(32,47)(34,50)(35,49)(37,39)(38,51)(42,54)(43,55)(45,48)(46,56)(52,53);(3,7)(5,8)(10,18)(15,19)(16,17)(23,24)(27,37)(28,40)(31,45)(33,49)(38,52)(41,54)(43,46)(47,50)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=f,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,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, 7112698);