#7205570

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1370 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)(25,26,30,34)(27,35,36,48)(28,39,40,32)(29,42,44,33)(31,43,45,38)(37,51,50,49)(41,47,55,54)(46,52,53,56);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(27,36)(31,45)(35,48)(37,50)(38,43)(46,53)(49,51)(52,56);(3,7)(5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,30,36)(26,31,34,45)(28,37,40,50)(29,43,44,38)(32,46,39,53)(33,48,42,35)(41,56,55,52)(47,51,54,49);(2,6)(3,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,32)(27,37)(29,41)(30,40)(31,46)(33,47)(34,39)(35,49)(36,50)(38,52)(42,54)(43,56)(44,55)(45,53)(48,51);(3,7)(5,8)(10,18)(15,19)(16,17)(23,24)(27,36)(28,40)(31,45)(32,39)(35,48)(38,43)(41,55)(47,54)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fi,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,e^c=ei,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, 7112766);