#193908

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	#2433	C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 101104);