#7552292

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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

1381 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=f,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7459488);