#7567041

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,22,25)(19,21,27,23)(20,24,30,32)(26,29,28,31);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,24)(21,25)(22,27)(26,29)(28,31)(30,32);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,21)(20,28)(22,27)(23,25)(24,31)(26,30)(29,32);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28);(2,6)(7,8)(10,14)(13,15)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,26,32,28)

Polycyclic presentation:

2^9|a^2=gi,c^2=fgi,d^2=fgi,e^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, 7474237);