#239863

order	512
nilpotency class	3
isoclinism	isoclinic class of #234079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(5,6,9,13)(7,10,14,18)(8,11,16,12)(15,17,20,19)(23,31)(24,25)(27,35)(28,29)(32,36)(33,34);(1,2)(3,4)(5,7,9,14)(6,10,13,18)(8,15,16,20)(11,17,12,19)(21,22,26,30)(23,32,33,35)(24,29,25,28)(27,31,36,34);(1,3,4,2)(6,11)(8,16)(10,17)(12,13)(15,20)(18,19)(21,23)(22,27)(24,34)(25,31)(26,33)(28,32)(29,35)(30,36);(7,14)(10,18)(15,20)(17,19)(23,33)(27,36)(31,34)(32,35)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dhi,f^b=fi,g^b=gi,d^c=di,f^c=fh,g^c=gi

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, 147059);