#7331447

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fg,d^2=fi,e^2=i,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^c=ei,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, 7238643);