#497463

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3416	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

964 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404659);