#352069

order	512
nilpotency class	3
isoclinism	isoclinic class of #352064
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

753 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)(5,6,10,8)(7,13,17,20)(9,18,14,12)(11,19,15,16)(21,22,27,32)(23,29,33,36)(24,28,34,35)(25,31,26,30);(5,7,14,19,10,17,9,16)(6,11,18,20,8,15,12,13)(21,23,26,34,27,33,25,24)(22,28,31,36,32,35,30,29);(5,8,10,6)(7,15,17,11)(9,18,14,12)(13,16,20,19)(21,24,25,33,27,34,26,23)(22,29,30,35,32,36,31,28);(2,4)(6,8)(11,15)(12,18)(13,20)(23,33)(24,34)(28,35)(29,36)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fh,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, 259265);