#7200843

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1418 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,36)(27,38,41,32)(28,34,43,50)(29,37,31,35)(33,45,49,42)(39,53,55,47)(40,48,44,51)(46,56,52,54);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,30,41)(26,32,36,38)(28,39,43,55)(29,44,31,40)(33,46,49,52)(34,47,50,53)(35,51,37,48)(42,56,45,54);(3,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28)(26,33)(27,39)(29,42)(30,43)(31,45)(32,46)(34,35)(36,49)(37,50)(38,52)(40,54)(41,55)(44,56)(47,51)(48,53);(2,6)(3,7)(11,19)(14,17)(18,24)(21,23)(25,28)(26,34)(27,39)(29,45)(30,43)(31,42)(32,47)(33,35)(36,50)(37,49)(38,53)(40,56)(41,55)(44,54)(46,51)(48,52);(3,7)(5,8)(10,16)(14,17)(15,22)(21,23)(25,29,30,31)(26,35,36,37)(27,40,41,44)(28,42,43,45)(32,48,38,51)(33,34,49,50)(39,54,55,56)(46,53,52,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7108039);