#7301741

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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

1423 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)(25,26,31,36)(27,32,40,48)(28,34,42,49)(29,35,30,44)(33,43,50,41)(37,46,52,55)(38,47,39,54)(45,53,56,51);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,31,40)(26,32,36,48)(28,37,42,52)(29,38,30,39)(33,45,50,56)(34,46,49,55)(35,47,44,54)(41,51,43,53);(1,3)(2,5)(4,8)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28)(26,33)(27,37)(29,41)(30,43)(31,42)(32,45)(34,44)(35,49)(36,50)(38,51)(39,53)(40,52)(46,54)(47,55)(48,56);(2,6)(3,8)(11,19)(14,17)(18,24)(21,23)(25,28)(26,34)(27,37)(29,43)(30,41)(31,42)(32,46)(33,44)(35,50)(36,49)(38,53)(39,51)(40,52)(45,54)(47,56)(48,55);(2,6)(5,7)(10,16)(14,17)(15,22)(21,23)(25,29,31,30)(26,35,36,44)(27,38,40,39)(28,41,42,43)(32,47,48,54)(33,49,50,34)(37,51,52,53)(45,55,56,46)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7208937);