#7531925

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1738 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,18,22,26)(19,23,28,38)(20,30,33,42)(21,34,36,25)(24,35,41,32)(27,43,31,48)(29,46,45,39)(37,47,40,44);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19)(18,23)(20,31)(21,29)(22,28)(24,40)(25,39)(26,38)(27,33)(30,48)(32,47)(34,46)(35,44)(36,45)(37,41)(42,43);(10,14)(13,15)(17,20,22,33)(18,24,26,41)(19,27,28,31)(21,35,36,32)(23,37,38,40)(25,42,34,30)(29,47,45,44)(39,48,46,43);(2,5)(6,8)(10,14)(13,15)(19,28)(23,38)(27,31)(29,45)(37,40)(39,46)(43,48)(44,47)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7439121);