#7200460

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	#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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^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=dfh,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, 7107656);