#7339803

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

1422 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:

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

Polycyclic presentation:

2^9|a^2=g,b^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.

Relationships

maximal quotient of

#72320049

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7246999);