#7181824

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1316 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=f,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7089020);