#6460655

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1614 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fgh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367851);