#352219

order	512
nilpotency class	3
isoclinism	isoclinic class of #352122
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

770 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:

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

Polycyclic presentation:

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

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, 259415);