#7544000

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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1573 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(18,19)(20,28)(25,27)(30,31);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,18)(19,22)(20,29)(21,25)(23,27)(24,31)(26,30)(28,32);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19,22,18)(20,26,28,24)(21,25,23,27)(29,31,32,30);(10,14)(13,15)(17,20,22,28)(18,24,19,26)(21,31,23,30)(25,32,27,29);(2,5)(6,8)(10,14)(13,15)(17,21,22,23)(18,25,19,27)(20,30,28,31)(24,29,26,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=fgi,d^2=i,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, 7451196);