#245287

order	512
nilpotency class	3
isoclinism	isoclinic class of #245282
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=h,b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,d^c=di,e^c=eh,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284205, #72284232, #72284327, …
maximal normal subgroup of	#72284205, #72284232, #72284327, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152483);