#374357

order	512
nilpotency class	3
isoclinism	isoclinic class of #196332
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fhi,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281553);