#352193

order	512
nilpotency class	3
isoclinism	isoclinic class of #352064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

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

745 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)(5,6,10,12)(7,14,16,20)(8,17,19,11)(9,18,15,13)(21,22,27,32)(23,33,34,29)(24,36,35,28)(25,31,26,30);(5,7,15,17,10,16,9,11)(6,8,18,20,12,19,13,14)(21,23,26,24,27,34,25,35)(22,28,31,29,32,36,30,33);(5,8,10,19)(6,11,12,17)(7,13,16,18)(9,14,15,20)(21,24,26,34,27,35,25,23)(22,29,31,36,32,33,30,28);(2,4)(6,12)(8,19)(13,18)(14,20)(23,34)(24,35)(28,36)(29,33)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=f,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=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, 259389);