#122198

order	512
nilpotency class	3
isoclinism	isoclinic class of #122196
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

639 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fh,d^2=i,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,g^b=ghi,d^c=di,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29394);