#6355556

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355552
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60968	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1649 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,13)(15,16)(18,22)(19,26)(20,29)(24,25)(27,28)(30,31);(1,3)(2,6)(4,11)(5,10)(7,13)(8,14)(9,15)(12,16)(17,18,20,22,21,24,29,25)(19,23,28,30,27,32,26,31);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,19)(18,23)(20,28)(21,27)(22,30)(24,32)(25,31)(26,29);(3,10)(6,14)(11,16)(13,15)(18,24)(19,27)(22,25)(26,28);(2,8)(6,14)(7,9)(13,15)(17,20,21,29)(18,25,24,22)(19,28,27,26)(23,31,32,30)

Polycyclic presentation:

2^9|b^2=f,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=di,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306813, #72306874, #72306925
maximal normal subgroup of	#72306813, #72306874, #72306925

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262752);