#497405

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×#3293	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1044 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)(9,27)(10,14)(13,19)(15,16)(18,36)(20,28)(21,34)(22,35)(23,31)(24,25)(29,30)(32,33);(5,6,11,16)(7,18,22,13)(8,23,26,10)(9,15,27,12)(14,25,31,24)(17,33,35,30)(19,28,36,34)(20,29,21,32);(5,7,11,22)(6,13,16,18)(8,19,26,36)(9,20,27,21)(10,28,23,34)(12,29,15,32)(14,17,31,35)(24,33,25,30);(5,8)(6,14)(7,19)(9,25)(10,12)(11,26)(13,17)(15,23)(16,31)(18,35)(20,30)(21,33)(22,36)(24,27)(28,29)(32,34)

Polycyclic presentation:

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

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, 404601);