#351461

order	512
nilpotency class	3
isoclinism	isoclinic class of #348318
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=efi,c^2=i,d^2=ei,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=ch,d^c=dh,g^d=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258657);