#321960

order	512
nilpotency class	3
isoclinism	isoclinic class of #321957
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	#19698	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

729 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:

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

Polycyclic presentation:

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

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, 229156);