#7566964

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
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1757 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fg,d^2=fgi,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,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, 7474160);