#116782

order	512
nilpotency class	3
isoclinism	isoclinic class of #10104
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

567 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=h,e^2=i,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,f^b=fhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23978);