#263392

order	512
nilpotency class	3
isoclinism	isoclinic class of #263387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62259	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

885 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fhi,d^c=di,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170588);