#322466

order	512
nilpotency class	3
isoclinism	isoclinic class of #322453
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

669 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)(3,13)(4,15)(5,8)(6,12)(7,11)(9,10)(14,16)(17,18,21,30,23,27,22,26)(19,25,20,24,29,32,28,31);(2,8,12,11)(3,14,10,15)(4,16)(9,13)(17,19,21,20,23,29,22,28)(18,24,26,25,27,31,30,32);(1,3)(2,9)(4,8)(5,14)(6,10)(7,15)(11,16)(12,13)(17,18,23,27)(19,24,29,31)(20,25,28,32)(21,26,22,30);(1,4,5,9,6,16,7,13)(2,10,11,15,12,3,8,14)(17,20,21,29,23,28,22,19)(18,25,26,31,27,32,30,24)

Polycyclic presentation:

2^9|a^2=fh,b^2=eg,c^2=i,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,f^c=fi,g^c=ghi

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, 229662);