#195763

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60782	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1157 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322122, #72322149, #72322204
maximal normal subgroup of	#72322122, #72322149, #72322204

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102959);