#7532924

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	#45265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1514 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)(11,15)(12,14)(13,16)(17,18,22,25)(19,24,30,32)(20,23,26,21)(27,29,31,28);(2,6)(7,8)(11,16)(13,15)(17,19)(18,24)(20,31)(21,29)(22,30)(23,28)(25,32)(26,27);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,22,26)(18,21,25,23)(19,27,30,31)(24,28,32,29);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(18,25)(19,28)(21,23)(24,31)(27,32)(29,30);(2,6)(7,8)(17,21,22,23)(18,26,25,20)(19,29,30,28)(24,27,32,31)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=f,e^2=i,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, 7440120);