#6355597

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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1671 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,15)(12,16)(18,22)(19,26)(20,29)(24,25)(27,28)(30,31);(1,3,5,13,6,12,4,11)(2,7,8,10,9,15,14,16)(17,18,20,22,21,24,29,25)(19,23,28,30,27,32,26,31);(1,2)(3,7)(4,14)(5,8)(6,9)(10,13)(11,16)(12,15)(17,19)(18,23)(20,28)(21,27)(22,30)(24,32)(25,31)(26,29);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,10,15,16)(18,24)(19,27)(22,25)(26,28);(3,12)(7,15)(10,16)(11,13)(17,20,21,29)(18,25,24,22)(19,28,27,26)(23,31,32,30)

Polycyclic presentation:

2^9|b^2=f,d^2=hi,e^2=i,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	#72338371, #72338434, #72338449, …
maximal normal subgroup of	#72338371, #72338434, #72338449, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262793);