#7198252

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

1301 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=fghi,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=dhi,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, 7105448);