#289434

order	512
nilpotency class	3
isoclinism	isoclinic class of #289395
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3151	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

561 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=efh,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=ghi,d^c=dh,f^c=fi,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196630);