#7201457

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

1434 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)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26,31,37)(27,39,43,33)(28,35,29,46)(30,38,32,36)(34,47,52,45)(40,54,41,49)(42,50,44,51)(48,55,53,56);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,31,43)(26,33,37,39)(28,40,29,41)(30,42,32,44)(34,48,52,53)(35,49,46,54)(36,50,38,51)(45,55,47,56);(3,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,31,29)(26,34,37,52)(27,40,43,41)(30,45,32,47)(33,48,39,53)(35,38,46,36)(42,55,44,56)(49,51,54,50);(2,6)(3,7)(11,19)(14,17)(18,24)(21,23)(25,29,31,28)(26,35,37,46)(27,41,43,40)(30,45,32,47)(33,49,39,54)(34,38,52,36)(42,55,44,56)(48,51,53,50);(3,7)(5,8)(10,16)(14,17)(15,22)(21,23)(25,30,31,32)(26,36,37,38)(27,42,43,44)(28,45,29,47)(33,50,39,51)(34,35,52,46)(40,55,41,56)(48,49,53,54)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,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, 7108653);