#196227

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60559	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1105 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,d^c=di,e^c=eh,f^c=fh,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103423);