#7339088

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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1381 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)(25,26,30,31)(27,35,38,52)(28,33,39,46)(29,43,44,34)(32,47,45,37)(36,53,40,56)(41,51,42,50)(48,54,49,55);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,17,18)(12,19,13,20)(16,23,22,24)(26,31)(28,39)(32,45)(34,43)(35,52)(36,40)(41,42)(54,55);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,29,37,30,38,44,47)(26,32,34,35,31,45,43,52)(28,40,42,54,39,36,41,55)(33,49,51,56,46,48,50,53);(2,6)(3,8)(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(25,28,30,39)(26,33,31,46)(27,36,38,40)(29,42,44,41)(32,48,45,49)(34,51,43,50)(35,53,52,56)(37,55,47,54);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29,30,44)(26,34,31,43)(27,37,38,47)(28,41,39,42)(32,35,45,52)(33,50,46,51)(36,54,40,55)(48,56,49,53)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fgi,d^2=f,e^2=f,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, 7246284);