#193763

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
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

1133 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100959);