#7198381

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1399 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=fghi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7105577);