#7339807

order	512
nilpotency class	3
isoclinism	isoclinic class of #7339802
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#48088
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1395 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=fg,f^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfh,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,h^d=hi,f^e=fi,g^e=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7247003);