#497285

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂×#525	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1117 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:

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

Polycyclic presentation:

2^9|a^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^b=de,g^c=gi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404481);