#7328594

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324436
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1326 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,29,31)(27,35,38,48)(28,33,40,44)(30,42,36,34)(32,45,43,39)(37,49,51,56)(41,53,50,47)(46,55,54,52);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(26,31)(28,40)(32,43)(34,42)(35,48)(37,51)(41,50)(52,55);(1,3)(2,5)(4,8)(6,7)(25,27,36,45,29,38,30,39)(26,32,42,48,31,43,34,35)(28,37,50,55,40,51,41,52)(33,46,53,56,44,54,47,49);(2,6)(3,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,33)(27,37)(29,40)(30,41)(31,44)(32,46)(34,47)(35,49)(36,50)(38,51)(39,52)(42,53)(43,54)(45,55)(48,56);(2,6)(5,7)(10,16)(14,17)(15,24)(22,23)(26,31)(28,40)(32,43)(34,42)(35,48)(37,51)(41,50)(52,55)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,d^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=dh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7235790);