#7124834

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 7032030);