#7589252

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	#25770	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1797 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)(11,15)(12,14)(13,16)(17,18,21,24)(19,22,25,29)(20,23,28,32)(26,30,27,31);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,28)(31,32);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(19,25)(22,29)(26,27)(30,31);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(19,26)(20,28)(22,30)(23,32)(25,27)(29,31);(1,4,5,3)(2,7,6,8)(10,14)(13,15)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,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, 7496448);