#7124710

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1395 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7031906);