#7525889

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

1585 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(18,23)(24,36)(25,37)(26,34)(30,40)(38,47)(39,45)(41,48);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18,21,23)(19,24,28,36)(20,30,31,40)(22,34,33,26)(25,35,37,32)(27,41,42,48)(29,45,44,39)(38,46,47,43);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,19,21,28)(18,24,23,36)(20,27,31,42)(22,29,33,44)(25,38,37,47)(26,39,34,45)(30,41,40,48)(32,43,35,46);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,20)(18,25)(19,27)(21,31)(22,35)(23,37)(24,38)(26,40)(28,42)(29,46)(30,34)(32,33)(36,47)(39,48)(41,45)(43,44);(2,6)(7,8)(19,28)(20,31)(24,36)(25,37)(29,44)(30,40)(32,35)(39,45)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7433085);