#7205318

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1419 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=f,d^2=f,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=ehi,e^c=ei,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, 7112514);