#6361848

order	512
nilpotency class	3
isoclinism	isoclinic class of #6361666
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1574 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)(3,13)(4,12)(5,10)(6,9)(7,11)(8,14)(15,16)(17,18,22,24)(19,26,27,32)(20,25,21,29)(23,30,31,28);(1,3,7,15)(2,8,11,5)(4,14,6,10)(9,13,12,16)(19,27)(23,31)(26,32)(28,30);(1,4,7,6)(2,9,11,12)(3,14,15,10)(5,16,8,13)(17,19,22,27)(18,23,24,31)(20,28,21,30)(25,32,29,26);(1,5)(2,10)(3,9)(4,16)(6,13)(7,8)(11,14)(12,15)(18,24)(23,31)(25,29)(26,32);(2,11)(3,15)(9,12)(10,14)(17,20,22,21)(18,25,24,29)(19,28,27,30)(23,32,31,26)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339713, #72339721, #72339809
maximal normal subgroup of	#72339713, #72339721, #72339809

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269044);