#7531422

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514196
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1830 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=f,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,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, 7438618);