#497401

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
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×#3290	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1055 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)(6,12)(7,17)(9,26)(10,14)(13,18)(15,16)(19,27)(20,33)(21,34)(22,31)(23,24)(28,29)(30,36)(32,35);(5,6,11,16)(7,13,21,30)(8,22,25,10)(9,15,26,12)(14,24,31,23)(17,29,34,35)(18,33,36,27)(19,32,20,28);(5,7)(6,13)(8,18)(9,19)(10,27)(11,21)(12,28)(14,17)(15,32)(16,30)(20,26)(22,33)(23,35)(24,29)(25,36)(31,34);(5,8)(6,14)(7,18)(9,24)(10,12)(11,25)(13,17)(15,22)(16,31)(19,29)(20,35)(21,36)(23,26)(27,28)(30,34)(32,33)

Polycyclic presentation:

2^9|a^2=g,b^2=i,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=de,e^b=ei,f^c=fh,h^c=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404597);