#286650

order	512
nilpotency class	3
isoclinism	isoclinic class of #281915
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	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

618 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)(3,6)(4,5)(7,8)(9,10,13,11,14,16,12,15)(17,18,20,24,21,23,25,29)(19,22,28,30,27,26,31,32);(2,5)(6,8)(9,11,12,10,14,15,13,16)(17,19,25,31,21,27,20,28)(18,22,29,32,23,26,24,30);(1,3,4,7)(2,6,5,8)(9,10,14,16)(11,13,15,12)(17,18)(19,26)(20,29)(21,23)(22,27)(24,25)(28,30)(31,32);(2,5)(6,8)(9,12,14,13)(10,11,16,15)(18,23)(19,27)(24,29)(28,31)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=ehi,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 193846);