#7567064

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

1563 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)(11,15,16,13);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18,22,26)(19,23,28,38)(20,30,32,42)(21,34,36,25)(24,35,41,33)(27,43,31,48)(29,46,45,39)(37,47,40,44);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19,22,28)(18,23,26,38)(20,31,32,27)(21,29,36,45)(24,40,41,37)(25,39,34,46)(30,48,42,43)(33,47,35,44);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20,22,32)(18,24,26,41)(19,27,28,31)(21,35,36,33)(23,37,38,40)(25,42,34,30)(29,47,45,44)(39,48,46,43);(2,6)(7,8)(10,14)(13,15)(19,28)(20,32)(23,38)(24,41)(29,45)(30,42)(33,35)(39,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fgi,d^2=fgi,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,e^d=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, 7474260);