#353002

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×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1000 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)(5,6,10,13)(7,14,12,9)(8,11,16,18)(15,20,19,17)(21,22,26,30)(23,28,31,36)(24,33,32,27)(25,35,34,29);(5,7,9,6,10,12,14,13)(8,15,17,11,16,19,20,18)(21,23,25,32,26,31,34,24)(22,27,29,28,30,33,35,36);(5,8,10,16)(6,11,13,18)(7,15,12,19)(9,17,14,20)(21,24,34,31,26,32,25,23)(22,28,35,27,30,36,29,33);(1,2,3,4)(8,16)(11,18)(15,19)(17,20)(23,31)(24,32)(27,33)(28,36)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,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, 260198);