#193878

order	512
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	#34963	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1476 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)(17,18,20,24,21,23,27,30)(19,22,25,29,26,28,31,32);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,19,20,25,21,26,27,31)(18,22,24,29,23,28,30,32);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,23)(22,28)(24,30)(29,32)

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,e^c=eh,g^c=gh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306481, #72306515, #72306518, …
maximal normal subgroup of	#72306481, #72306515, #72306518, …
maximal quotient of	#72343161, #72343189

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101074);