#7338045

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324564
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	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1366 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)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26)(27,36)(28,39)(29,34)(30,35)(31,32)(33,42)(37,49)(38,50)(40,55)(41,46)(43,47)(44,56)(45,52)(48,54)(51,53);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,31)(26,32,35,36)(28,37,41,42)(29,43,45,38)(33,46,49,39)(34,50,52,47)(40,54,44,53)(48,55,51,56);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,27,37,30,41,31,42)(26,33,32,46,35,49,36,39)(29,44,38,54,45,40,43,53)(34,51,47,55,52,48,50,56);(2,6)(3,8)(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(25,29,30,45)(26,34,35,52)(27,38,31,43)(28,40,41,44)(32,47,36,50)(33,48,49,51)(37,53,42,54)(39,55,46,56);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,35)(29,45)(32,36)(33,49)(38,43)(39,46)(40,44)(53,54)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7245241);