#7203843

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1403 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,31,35)(27,36,39,46)(28,33,40,48)(29,34,30,43)(32,44,47,38)(37,51,52,56)(41,49,42,50)(45,53,55,54);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(28,40)(33,48)(37,52)(41,42)(45,55)(49,50)(51,56)(53,54);(3,7)(5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,31,39)(26,32,35,47)(28,37,40,52)(29,38,30,44)(33,45,48,55)(34,46,43,36)(41,53,42,54)(49,51,50,56);(2,6)(3,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,37)(29,42)(30,41)(31,40)(32,45)(34,50)(35,48)(36,51)(38,54)(39,52)(43,49)(44,53)(46,56)(47,55);(3,7)(5,8)(10,18)(15,19)(16,17)(23,24)(25,29,31,30)(26,34,35,43)(27,38,39,44)(28,41,40,42)(32,46,47,36)(33,49,48,50)(37,53,52,54)(45,51,55,56)

Polycyclic presentation:

2^9|a^2=i,c^2=fi,d^2=f,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=eh,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, 7111039);