#195497

order	512
nilpotency class	3
isoclinism	isoclinic class of #18524
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1096 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)(3,12,14,16)(4,13,15,8)(5,9)(6,10)(7,11)(17,18)(19,27)(20,26)(21,23)(22,25)(24,28)(29,32)(30,31);(1,3,5,13,7,14,6,8)(2,4,9,12,11,15,10,16)(18,23)(24,31)(25,26)(27,32);(1,4)(2,8)(3,9)(5,12)(6,16)(7,15)(10,14)(11,13)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,27)(26,32);(1,5,7,6)(2,9,11,10)(3,13,14,8)(4,12,15,16)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,27,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=dhi,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102693);