#7234264

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207830
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1191 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fghi,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=di,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, 7141460);