#7186458

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

1195 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)(12,20)(15,21)(19,24)(22,23)(25,26,31,35)(27,32,30,44)(28,38,41,47)(29,34,43,50)(33,36,48,40)(37,45,42,49)(39,52,54,56)(46,51,55,53);(2,5)(3,7)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27,31,30)(26,32,35,44)(28,36,41,40)(29,37,43,42)(33,38,48,47)(34,45,50,49)(39,51,54,53)(46,52,55,56);(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,28,27,36,31,41,30,40)(26,33,44,47,35,48,32,38)(29,39,37,51,43,54,42,53)(34,46,49,56,50,55,45,52);(1,3,4,7)(2,6,5,8)(11,18)(14,17)(19,24)(22,23)(25,29,31,43)(26,34,35,50)(27,37,30,42)(28,39,41,54)(32,45,44,49)(33,46,48,55)(36,51,40,53)(38,52,47,56);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(25,27,31,30)(26,32,35,44)(28,40,41,36)(29,42,43,37)(33,47,48,38)(34,49,50,45)(39,51,54,53)(46,52,55,56)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gi,d^2=fi,e^2=hi,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,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7093654);