#7338065

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1359 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)(25,26,30,34)(27,35,38,31)(28,39,41,52)(29,33,44,51)(32,46,48,36)(37,49,42,45)(40,55,43,56)(47,54,50,53);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,38)(26,31,34,35)(28,36,41,46)(29,42,44,37)(32,39,48,52)(33,49,51,45)(40,54,43,53)(47,56,50,55);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,27,36,30,41,38,46)(26,32,31,39,34,48,35,52)(29,43,37,54,44,40,42,53)(33,50,45,56,51,47,49,55);(2,6)(3,8)(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(25,29,30,44)(26,33,34,51)(27,37,38,42)(28,40,41,43)(31,45,35,49)(32,47,48,50)(36,53,46,54)(39,55,52,56);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,34)(29,44)(31,35)(32,48)(37,42)(39,52)(40,43)(53,54)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,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, 7245261);