#7306784

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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

1303 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7213980);