#353012

order	512
nilpotency class	3
isoclinism	isoclinic class of #352064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

969 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fi,d^2=g,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=di,d^c=dh

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, 260208);