#236926

order	512
nilpotency class	3
isoclinism	isoclinic class of #10144
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1228 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,21,26)(19,27,30,32)(20,24,22,25)(23,28,31,29);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19,21,30)(18,23,26,31)(20,28,22,29)(24,27,25,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,18)(19,27)(20,24)(21,26)(22,25)(23,29)(28,31)(30,32);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,21,22)(18,24,26,25)(19,28,30,29)(23,27,31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,g^b=gh,e^c=ei,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, 144122);