#7589776

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	#24257	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1788 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fgi,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,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.

Relationships

maximal subgroup of	#72257376, #72257520, #72264133
maximal normal subgroup of	#72257376, #72257520, #72264133
maximal quotient of	#72308419

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7496972);