#322457

order	512
nilpotency class	3
isoclinism	isoclinic class of #322453
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

667 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229653);