#321997

order	512
nilpotency class	3
isoclinism	isoclinic class of #321983
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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

657 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,d^2=ei,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,f^c=fi,g^c=gh

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, 229193);