#7051556

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1223 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)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,35)(28,33)(29,31)(30,43)(32,38)(34,42)(36,50)(37,49)(39,45)(40,54)(41,56)(44,51)(46,53)(47,55)(48,52);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,31)(27,36)(28,39)(34,43)(38,51)(41,52)(46,47)(49,54);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,30,38,29,36,42,51)(26,32,34,35,31,44,43,50)(28,40,41,55,39,37,52,53)(33,47,48,54,45,46,56,49);(1,2,5,4)(3,7,8,6)(12,20)(15,24)(17,19)(21,23)(25,28,29,39)(26,33,31,45)(27,37,36,40)(30,41,42,52)(32,46,44,47)(34,48,43,56)(35,49,50,54)(38,53,51,55);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(26,31)(27,36)(28,39)(34,43)(38,51)(41,52)(46,47)(49,54)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=g,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 6958752);