#321927

order	512
nilpotency class	3
isoclinism	isoclinic class of #321901
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	#10280	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

771 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229123);