#374398

order	512
nilpotency class	3
isoclinism	isoclinic class of #196266
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,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, 281594);