#196220

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1111 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103416);