#352998

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,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, 260194);