#7533525

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×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1549 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,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, 7440721);