#246916

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

996 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,c^b=cg,d^b=di,e^b=eh,f^b=fi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154112);