#289427

order	512
nilpotency class	3
isoclinism	isoclinic class of #289395
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

531 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,6,10,7,11,5,8)(3,4,12,15,13,14,16,9)(17,18,23,29)(19,26,20,32)(21,35,37,48)(22,39,24,28)(25,34,41,31)(27,38,45,36)(30,43,33,40)(42,47,44,46);(2,8,11,10)(3,12,13,16)(4,14)(9,15)(17,19,24,34,23,20,22,31)(18,25,39,32,29,41,28,26)(21,33,38,46,37,30,36,47)(27,43,48,42,45,40,35,44);(1,3,7,13)(2,9,11,15)(4,10,14,8)(5,12,6,16)(17,20)(18,26)(19,23)(21,30)(22,34)(24,31)(25,39)(27,44)(28,41)(29,32)(33,37)(35,40)(36,46)(38,47)(42,45)(43,48);(1,4,6,15,7,14,5,9)(2,3,8,16,11,13,10,12)(17,21,22,36,23,37,24,38)(18,27,28,35,29,45,39,48)(19,30,31,46,20,33,34,47)(25,40,26,42,41,43,32,44)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,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,d^b=dh,g^b=ghi,d^c=dh,f^c=fi,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196623);